Astrophysics : Decoding the Cosmos

AstrophysicsDecoding the Cosmos

Judith A. Irwin

Queen’s University, Kingston, Canada

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AstrophysicsDecoding the Cosmos

Judith A. Irwin

Queen’s University, Kingston, Canada

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Contents

Preface xiii

Acknowledgments xv

Introduction xvii

Appendix: dimensions, units and equations xxii

PART I THE SIGNAL OBSERVED

1 Defining the signal 3

1.1 The power of light – luminosity and spectral power 3

1.2 Light through a surface – flux and flux density 7

1.3 The brightness of light – intensity and specific intensity 9

1.4 Light from all angles – energy density and mean intensity 13

1.5 How light pushes – radiation pressure 16

1.6 The human perception of light – magnitudes 191.6.1 Apparent magnitude 19

1.6.2 Absolute magnitude 22

1.6.3 The colour index, bolometric correction, and HR diagram 23

1.6.4 Magnitudes beyond stars 24

1.7 Light aligned – polarization 25

Problems 25

2 Measuring the signal 29

2.1 Spectral filters and the panchromatic universe 29

2.2 Catching the signal – the telescope 322.2.1 Collecting and focussing the signal 34

2.2.2 Detecting the signal 36

2.2.3 Field of view and pixel resolution 37

2.2.4 Diffraction and diffraction-limited resolution 38

2.3 The Corrupted signal – the atmosphere 412.3.1 Atmospheric refraction 42

2.3.2 Seeing 43

2.3.3 Adaptive optics 45

2.3.4 Scintillation 48

2.3.5 Atmospheric reddening 48

2.4 Processing the signal 492.4.1 Correcting the signal 49

2.4.2 Calibrating the signal 50

2.5 Analysing the signal 50

2.6 Visualizing the signal 52

Problems 55

Appendix: refraction in the Earth’s atmosphere 58

PART II MATTER AND RADIATION ESSENTIALS

3 Matter essentials 65

3.1 The Big Bang 65

3.2 Dark and light matter 66

3.3 Abundances of the elements 703.3.1 Primordial abundance 70

3.3.2 Stellar evolution and ISM enrichment 70

3.3.3 Supernovae and explosive nucleosynthesis 75

3.3.4 Abundances in the Milky Way, its star formation history

and the IMF 77

3.4 The gaseous universe 823.4.1 Kinetic temperature and the Maxwell–Boltzmann

velocity distribution 84

3.4.2 The ideal gas 88

3.4.3 The mean free path and collision rate 89

3.4.4 Statistical equilibrium, thermodynamic equilibrium, and LTE 93

3.4.5 Excitation and the Boltzmann Equation 96

3.4.6 Ionization and the Saha Equation 100

3.4.7 Probing the gas 101

3.5 The dusty Universe 1043.5.1 Observational effects of dust 105

3.5.2 Structure and composition of dust 109

3.5.3 The origin of dust 111

3.6 Cosmic rays 1123.6.1 Cosmic ray composition 112

3.6.2 The cosmic ray energy spectrum 113

3.6.3 The origin of cosmic rays 117

Problems 118

Appendix: the electron/proton ratio in cosmic rays 121

4 Radiation essentials 123

4.1 Black body radiation 1234.1.1 The brightness temperature 127

4.1.2 The Rayleigh–Jeans Law and Wien’s Law 129

4.1.3 Wien’s Displacement Law and stellar colour 130

viii CONTENTS

4.1.4 The Stefan–Boltzmann Law, stellar luminosity

and the HR diagram 132

4.1.5 Energy density and pressure in stars 134

4.2 Grey bodies and planetary temperatures 1344.2.1 The equilibrium temperature of a grey body 136

4.2.2 Direct detection of extrasolar planets 140

Problems 143

Appendix: derivation of the Planck function 146

4.A.1 The statistical weight 146

4.A.2 The mean energy per state 147

4.A.3 The specific energy density and specific intensity 148

PART III THE SIGNAL PERTURBED

5 The interaction of light with matter 153

5.1 The photon redirected – scattering 1545.1.1 Elastic scattering 157

5.1.2 Inelastic scattering 165

5.2 The photon lost – absorption 1685.2.1 Particle kinetic energy – heating 168

5.2.2 Change of state – ionization and the Stromgren sphere 169

5.3 The wavefront redirected – refraction 172

5.4 Quantifying opacity and transparency 1755.4.1 Total opacity and the optical depth 175

5.4.2 Dynamics of opacity – pulsation and stellar winds 178

5.5 The opacity of dust – extinction 182

Problems 183

6 The signal transferred 187

6.1 Types of energy transfer 187

6.2 The equation of transfer 189

6.3 Solutions to the equation of transfer 1916.3.1 Case A: no cloud 191

6.3.2 Case B: absorbing, but not emitting cloud 192

6.3.3 Case C: emitting, but not absorbing cloud 192

6.3.4 Case D: cloud in thermodynamic equilibrium (TE) 193

6.3.5 Case E: emitting and absorbing cloud 193

6.3.6 Case F: emitting and absorbing cloud in LTE 194

6.4 Implications of the LTE solution 1956.4.1 Implications for temperature 195

6.4.2 Observability of emission and absorption lines 196

6.4.3 Determining temperature and optical depth of HI clouds 200

Problems 204

7 The interaction of light with space 207

7.1 Space and time 207

7.2 Redshifts and blueshifts 2107.2.1 The Doppler shift – deciphering dynamics 210

7.2.2 The expansion redshift 217

7.2.3 The gravitational redshift 219

CONTENTS ix

7.3 Gravitational refraction 2207.3.1 Geometry and mass of a gravitational lens 220

7.3.2 Microlensing – MACHOs and planets 225

7.3.3 Cosmological distances with gravitational lenses 226

7.4 Time variability and source size 227

Problems 228

PART IV THE SIGNAL EMITTED

8 Continuum emission 235

8.1 Characteristics of continuum emission – thermal

and non-thermal 236

8.2 Bremsstrahlung (free–free) emission 2378.2.1 The thermal Bremsstrahlung spectrum 237

8.2.2 Radio emission from HII and other ionized regions 242

8.2.3 X-ray emission from hot diffuse gas 245

8.3 Free–bound (recombination) emission 251

8.4 Two-photon emission 253

8.5 Synchrotron (and cyclotron) radiation 2558.5.1 Cyclotron radiation – planets to pulsars 258

8.5.2 The synchrotron spectrum 262

8.5.3 Determining synchrotron source properties 268

8.5.4 Synchrotron sources – spurs, bubbles, jets, lobes and relics 270

8.6 Inverse Compton radiation 273

Problems 276

9 Line emission 279

9.1 The richness of the spectrum – radio waves to gamma rays 2809.1.1 Electronic transitions – optical and UV lines 280

9.1.2 Rotational and vibrational transitions – molecules, IR

and mm-wave spectra 281

9.1.3 Nuclear transitions – g-rays and high energy events 286

9.2 The line strengths, thermalization, and the critical

gas density 288

9.3 Line broadening 2909.3.1 Doppler broadening and temperature diagnostics 291

9.3.2 Pressure broadening 295

9.4 Probing physical conditions via electronic transitions 2969.4.1 Radio recombination lines 297

9.4.2 Optical recombination lines 302

9.4.3 The 21 cm line of hydrogen 307

9.5 Probing physical conditions via molecular transitions 3119.5.1 The CO molecule 311

Problems 313

PART V THE SIGNAL DECODED

10 Forensic astronomy 319

10.1 Complex spectra 319

x CONTENTS

10.1.1 Isolating the signal 319

10.1.2 Modelling the signal 321

10.2 Case studies – the active, the young, and the old 32510.2.1 Case study 1: the Galactic Centre 326

10.2.2 Case study 2: the Cygnus star forming complex 329

10.2.3 Case study 3: the globular cluster, NGC 6397 331

10.3 The messenger and the message 335

Problems 336

Appendix A: Mathematical and geometrical relations 339

A.1 Taylor series 339

A.2 Binomial expansion 339

A.3 Exponential expansion 340

A.4 Convolution 340

A.5 Properties of the ellipse 340

Appendix B: Astronomical geometry 343

B.1 One-dimensional and two-dimensional angles 343

B.2 Solid angle and the spherical coordinate system 344

Appendix C: The hydrogen atom 347

C.1 The hydrogen spectrum and principal quantum number 347

C.2 Quantum numbers, degeneracy, and statistical weight 352

C.3 Fine structure and the Zeeman effect 353

C.4 The l 21 cm line of neutral hydrogen 354

Appendix D: Scattering processes 357

D.1 Elastic, or coherent scattering 358D.1.1 Scattering from free electrons – Thomson scattering 358

D.1.2 Scattering from bound electrons I: the oscillator model 359

D.1.3 Scattering from bound electrons II: quantum mechanics 362

D.1.4 Scattering from bound electrons III: resonance scattering

and the natural line shape 363

D.1.5 Scattering from bound electrons IV: Rayleigh scattering 366

D.2 Inelastic scattering – Compton scattering from free electrons 367

D.3 Scattering by dust 369

Appendix E: Plasmas, the plasma frequency, and plasma waves 373

Appendix F: The Hubble relation and the expanding Universe 377

F.1 Kinematics of the Universe 377

F.2 Dynamics of the Universe 383

F.3 Kinematics, dynamics and high redshifts 386

Appendix G: Tables and figures 389

References 401

Index 407

CONTENTS xi

Preface

Like many textbooks, this one originated from lectures delivered over a number of

years to undergraduate students at my home institution – Queen’s University in

Kingston, Canada. These students had already taken a first year (or two) of physics

and one introductory astronomy course. Thus, this book is aimed at an intermediate

level and is meant to be a stepping stone to more sophisticated and focussed courses,

such as stellar structure, physics of the interstellar medium, cosmology, or others. The

text may also be of some help to beginning graduate students with little background in

astronomy or those who would like to see how physics is applied, in a practical way, to

astronomical objects.

The astronomy prerequisite is helpful, but perhaps not required for students at a more

senior level, since I make few assumptions as to prior knowledge of astronomy. I do

assume that students have some familiarity with celestial coordinate systems (e.g.

Right Ascension and Declination or others), although it is not necessary to know the

details of such systems to understand the material in this text. I also do not provide any

explanation as to how astronomical distances are obtained. Distances are simply

assumed to be known or not known, as the case may be. I provide some figures that

are meant to help with ‘astronomical geography’, but a basic knowledge of astronom-

ical scales would also be an asset, such as understanding that the Solar System is tiny in

comparison to the Galaxy and rotates about the Galactic centre.

As for approach, I had several goals in mind while organizing this material. First of

all, I did not want to make the book too ‘object-oriented’. That is, I did not want to write

a great deal of descriptive material about specific astronomical objects. For one thing,

astronomy is such a fast-paced field that these descriptions could easily and quickly

become out of date. And for another, in the age of the internet, it is very easy for

students to quickly download any number of descriptions of various astronomical

objects at their leisure. What is more difficult is finding the thread of physics that links

these objects, and it is this that I wanted to address.

Another goal was to keep the book practical, focussing on how we obtain informa-

tion about our Universe from the signal that we actually detect. In the process, many

equations are presented. While this might be a little intimidating to some students, the

point should be made that the equations are our ‘tools of the trade’. Without these tools,

we would be quite helpless, but with them, we have access to the secrets that

astronomical signals bring to us. With the increasing availability of computer algebra

or other software, there is no longer any need to be encumbered by mathematics.

Nevertheless, I have kept problems that require computer-based solutions to a mini-

mum in this text, and have tried to include problems over a range of difficulty.

Astrophysics – Decoding the Cosmos will maintain a website at

http://decoding.phy.queensu.ca. A Solutions manual to the problems is also available.

I invite readers to visit the website and submit some problems of their own so that these

can be shared with others. It is my sincere desire that this book will be a useful

stepping-stone for students of astrophysics and, more importantly, that it may play a

small part in illuminating this most remarkable and marvelous Universe that we live in.

Judith A. IrwinKingston, Ontario

October 2006

xiv PREFACE

Acknowledgments

There are some important people that I feel honoured to thank for their help, patience,

and critical assistance with this book. First of all, to the many people who generously

allowed me to use their images and diagrams, I am very grateful. Astronomy is a visual

science and the impact of these images cannot be overstated.

Thanks to the students, past and present, of Physics 315 for their questions and

suggestions. I have more than once had to make corrections as a result of these queries

and appreciate the keen and lively intelligence that these students have shown.

Thanks to Jeff Ross for his assistance with some of the ‘nuts and bolts’ of the

references. And special thanks to Kris Marble and Aimee Burrows for their many

contributions, working steadfastly through problems, and offering scientific expertise.

With much gratitude, I wish to acknowledge those individuals who have read

sections or chapters of this book and offered constructive criticism – Terry Bridges,

Diego Casadei, Mark Chen, Roland Diehl, Martin Duncan, David Gray, Jim Peebles,

Ernie Seaquist, David Turner and Larry Widrow.

To my dear children, Alex and Irene, thank you for your understanding and good

cheer when mom was working behind a closed door yet again, and also thanks to the

encouragement of friends – Joanne, Wendy, and Carolyn.

My tenderest thanks to my husband, Richard Henriksen, who not only suggested the

title to this book, but also read through and critiqued the entire manuscript. For

patience, endurance, and gentle encouragement, I thank you. It would not have been

accomplished without you.

J.A.I.

Introduction

Knowledge of our Universe has grown explosively over the past few decades, with

discoveries of cometary objects in the far reaches of our Solar System, new-found

planets around other stars, detections of powerful gamma ray bursts, galaxies in the

process of formation in the infant Universe, and evidence of a mysterious force that

appears to be accelerating the expansion of the Universe. From exotic black holes to the

microwave background, the modern understanding of our larger cosmological home

could barely be imagined just a generation ago. Headlines exclaim astonishing proper-

ties for astronomical objects – stars with densities equivalent to the mass of the sun

compressed to the size of a city, million degree gas temperatures, energy sources of

incredible power, and luminosities as great as an entire galaxy from a single dying star.

How could we possibly have reached these conclusions? How can we dare to

describe objects so inconceivably distant that the only influence they have on our lives

is through our very astonishment at their existence? With the exception of a few

footprints on the Moon, no human being has ever ventured beyond the confines of the

Earth. No spaceprobe has ever approached a star other than the Sun, let alone returned

with material evidence. Yet we continue to amass information about our Universe and,

indeed, to believe it. How?

In contrast to our attempts to reach into the Universe with space probes and radio

beacons, the natural Universe itself has been continuously and quite effectively reach-

ing us. The Earth has been bombarded with astronomical information in its various

forms and in its own language. The forms that we think we understand are those of

matter and radiation. Our challenge, in the absence of an ability to travel amongst the

stars, is to find the best ways to detect and decipher such communications.

What astronomical matter is reaching us? The high energy subatomic particles and

nuclei which make up cosmic rays that continuously bombard the earth (Sect. 3.6) as

well as an influx of meteoritic dust, occasional meteorites and those rare objects that

are large enough to create impact craters. Such incoming matter provides us with

information on a variety of astronomical sources, including our Sun, our Solar System,

supernovae in the Galaxy1, and other more mysterious sources of the highest energy

cosmic rays. The striking contrast in size and effect between such particulate matter is

illustrated in Figure I.1.

Radiation refers to electromagnetic radiation (or just ‘light’) over all wavebands

from the radio to the gamma-ray part of the spectrum (Table G.6). Electromagnetic

radiation can be described as a wave and identified by its wavelength, l or its

frequency, n. However, it can also be thought of as a massless particle called a photon

which has a particular energy, Eph. This energy can be expressed in terms of

wavelength or frequency, Eph ¼ h c=l ¼ h n, where h is Planck’s constant. The

wavelengths, frequencies and photon energies of various wavebands are given in

Table G.6. The wave–particle duality of light is a deep issue in physics and related

via the concept of probability. To quote Max Born, ‘the wave and corpuscular

descriptions are only to be regarded as complementary ways of viewing one and

the same objective process, a process which only in definite limiting cases admits of

complete pictorial interpretation’ (Ref. [23]). Although, in principle, it may be

possible to understand a physical process involving light from both points of view,

there are some problems that are more easily addressed by one approach rather than

another. For example, it is often more straightforward to consider waves when

dealing with an interaction between light and an object that is small in comparison

to the wavelength, and to consider photons when dealing with an interaction between

light and an object that is large in comparison to the equivalent photon wavelength,

l ¼ h c=Eph. Given this wave-particle duality, in this text we will apply whatever

form is most useful for the task at hand. Some helpful expressions relating various

properties of electromagnetic radiation are provided in Table I.1 and a diagram

illustrating the wave nature of light is shown in Figure I.2.

Figure I.1 (a) Early photograph of cosmic ray tracks in a cloud chamber. (Ref. [30]) (b) The1.8 km diameter Lonar meteorite crater in India

1When ‘Galaxy’ is written with a capital G, it refers to our own Milky Way galaxy.

xviii INTRODUCTION

Table I.1. Useful expressions relevant to light, matter, and fieldsa

Meaning Equation

Relation between wavelength and frequency c ¼ l n

Lorentz factor g ¼ 1ffiffiffiffiffiffiffiffiffi1 v2

c2

pEnergy of a photon E ¼ h n ¼ h c

l

Equivalent mass of a photon m ¼ E=c2

Equivalent wavelength of a mass (de Broglie wavelength) l ¼ h=ðm vÞMomentum of a photon p ¼ E=c

Momentum of a particle p ¼ g m v

Snell’s law of refractionb n1 sin u1 ¼ n2 sin u2

Index of refractionc n ¼ c=v

Doppler shiftdDl

l0

¼ lobs l0

l0

¼1 þ vr

c

1 vrc

1=2

1

Dn

n0

¼ nobs n0

n0

¼1 vr

c

1 þ vrc

1=2

1

Dl

l0

vrc;Dn

n0

vrc

; ðv cÞ

Electric field vector ~E ¼~F=q

Electric field vector of a wavee ~E ¼ ~E0 cos½2p zl n t

þ Df

Magnetic field vector of a wavee ~B ¼ ~B0 cos½2p zl n t

þ Df

Poynting fluxf ~S ¼ c4p

~E ~B

Time averaged Poynting fluxf hSi ¼ c

8pE0 B0

Energy density of a magnetic fieldg uB ¼ B2

8p

Lorentz forceh ~F ¼ q ð~E þ ~vc~BÞ

Electric field magnitude in a parallel-plate capacitori E ¼ ð4pN eÞ=AElectric dipole momentj ~p ¼ q~r

Larmor’s formula for powerk P ¼ dE

dt¼ 2 q2

3 c3j€~rj2

Heisenberg Uncertainty Principlel D xD px ¼h

2p

DED t ¼ h

2p

Universal law of gravitationm FG ¼ G M mr2

Centripetal forcen Fc ¼m v2

r

a See Table A.3 for a list of symbols if they are not defined here.bIf an incoming ray is travelling from medium 1 with index of refraction, n1, into medium 2 with index of refraction, n2, then u1 is the

angle between the incoming ray and the normal to the surface dividing the two media and u2 is the angle between the outgoing ray and the

normal to the surface.c v is the speed of light in the medium (the phase velocity) and c is the speed of light in a vacuum. Note that the index of refraction may

also be expressed as a complex number whose real part is given by this equation and whose imaginery part corresponds to an absorbed

component of light. See Appendix D.3 for an example.d l0 is the wavelength of the light in the source’s reference frame (the ‘true’ wavelength), lobs is the wavelength in the observer’s reference

frame (the measured wavelength), and vr is the relative radial velocity between the source and the observer. vr is taken to be positive if the

source and observer are receding with respect to each other and negative if the source and observer are approaching each other.e The wave is propagating in the z direction and Df is an arbitrary phase shift. The magnetic field strength, H, is given by B ¼ H m where

m is the permeability of the substance through which the wave is travelling (unitless in the cgs system). For em radiation in a vacuum

(assumed here and throughout), this becomes B ¼ H since the permeability of free space takes the value, 1, in cgs units. Thus B is often

stated as the magnetic field strength, rather than the magnetic flux density and is commonly expressed in units of Gauss. In cgs units,

E ðdyn esu1Þ ¼ B (Gauss).f Energy flux carried by the wave in the direction of propagation. The cgs units are erg s1 cm2. The time-averaged value is over one

cycle.g uB has cgs units of erg cm3 or dyn cm2 (see Table A.2.).h Force on a charge, q with velocity, ~v, by an electric field, ~E and magnetic field, ~B.i Here N e is the charge on a plate and A is its area. In SI units, this equation would be E ¼ s=e0, where s is the charge per unit area on a

plate and e0 is the permittivity of free space (where we assume that free space is between the plates). In cgs units, 4p e0 ¼ 1.j~r is the separation between the two charges of the dipole and q is the strength of one of the charges. The direction is negative to positive.k Power emitted by a non-relativistic particle of charge, q, that is accelerating at a rate, €~r.l One cannot know the position and momentum (x; p) or the energy and time (E; t) of a particle or photon to arbitrary accuracy.mForce between two masses, M and m, a distance, r, apart.n Force on an object of mass, m, moving at speed, v, in a circular path of radius, r.

If we now ask which of these information bearers provides us with most of our

current knowledge of the universe, the answer is undoubtedly electromagnetic radia-

tion, with cosmic rays a distant second. The world’s astronomical volumes would be

empty indeed were it not for an understanding of radiation and its interaction with

matter. The radiation may come directly from the object of interest, as when sunlight

travels straight to us, or it may be indirect, such as when we infer the presence of a black

hole by the X-rays emitted from a surrounding accretion disk. Even when we send out

exploratory astronomical probes, we still rely on man-made radiation to transfer the

images and data back to earth.

This volume is thus largely devoted to understanding radiative processes and how

such an understanding informs us about our Universe and the astronomical objects that

inhabit it. It is interesting that, in order to understand the largest and grandest objects in

the Universe, we must very often appeal to microscopic physics, for it is on such scales

that the radiation is actually being generated and it is on such scales that matter

interacts with it. We also focus on the ‘how’ of astronomy. How do we know the

temperature of that asteroid? How do we find the speed of that star? How do we know

the density of that interstellar cloud? How can we find the energy of that distant quasar?

The answers are hidden in the radiation that they emit and, earthbound, we have at least

a few keys to unlock their secrets. We can truly think of the detected signal as a coded

message. To understand the message requires careful decoding.

In the future, there may be new, exotic and perhaps unexpected ways in which we can

gather information about our Universe. Already, we are seeing a trend towards such

diversification. The decades-old ‘Solar neutrino problem’ has finally been solved from

the vantage point of an underground mine in the Canadian shield (Figure I.3.a).

Creative experiments are underway to detect the elusive dark matter that is believed

to make up most of the mass of the Universe but whose nature is unknown (Sect. 3.2).

Enormous international efforts are also underway to detect gravitational waves, weak

perturbations in space-time predicted by Einstein’s General Theory of Relativity

λE

E0

B0y

z

xS

B

Figure I.2 Illustration of an electromagnetic wave showing the electric field and magnetic fieldperpendicular to each other and perpendicular to the direction of wave propagation which is in thex direction. The wavelength is denoted by l.

xx INTRODUCTION

(Figure I.3.b). Our astronomical observatories are no longer restricted to lonely

mountain tops. They are deep underground, in space, and in the laboratory. Thus,

‘decoding the cosmos’ is an on-going and evolving process. Here are a few steps along

the path.

Problems

0.1 Calculate the repulsive force of an electron on another electron a distance, 1 m, away

in cgs units using Eq. 0.A.1 and in SI units using the equation,

F ¼ kq1 q2

r2ðI:1Þ

where the charge on an electron, in SI units, is 1:6 1019 Coulombs (C) and the constant,

k ¼ 8:988 109 N m2 C2. Verify that the result is the same in the two systems.

0.2 Verify that the following equations have matching units on both sides of the equals

sign: Eq. D.2 (the expression that includes the mass of the electron, me), Eq. 9.8 and

Eq. F.19.

Figure I.3 (a) The acrylic tank of the Sudbury Neutrino Observatory (SNO) looks like a coiledsnakeskin in this fish-eye view from the bottom of the tank before the bottom-most panel ofphotomultiplier tubes was installed. The Canadian-led SNO project determined that neutrinoschange ‘flavour’ en route from the Sun’s interior, thus answering why previous experiments haddetected fewer solar neutrinos than theory predicted. (Photo credit: Ernest Orlando, LawrenceBerkeley National Laboratory. Courtesy A. Mc Donald, SNO) (b) Aerial photo of the ’L’ shaped LaserInterferometer Gravitational-Wave Observatory (LIGO) in Livingston, Louisiana, of length 4 km oneach arm. Together with its sister observatory in Hanford, Washington, these interferometers maydetect gravitational waves, tiny distortions in space–time produced by accelerating masses.(Reproduced courtesy of LIGO, Livingston, Lousiana)

INTRODUCTION xxi

0.3 Verify that the equation for the Ideal Gas Law (Table 0.A.2) is equivalent to

PV ¼ N RT, where N is the number of moles and R is the molar gas constant (see

Table G.2).

Appendix

Dimensions, Units and Equations

When [chemists] had unscrambled the difficulties caused by the fact that chemists and engineers

use different units . . . they found that their strength predictions were not only frequently a

thousandfold in error but bore no consistent relationship with experiment at all. After this they

were inclined to give the whole thing up and to claim that the subject was of no interest or

importance anyway.

–The New Science of Strong Materials, by J. E. Gordon

The centimetre-gram-second (cgs) system of units is widely used by astronomers

internationally and is the system adopted in this text. A summary of the units is given in

Table 0.A.1 as well as corresponding conversions to Systeme Internationale (SI). If an

equation is given without units, the cgs system is understood. The same symbols are

generally used in both systems (Table 0.A.3) and SI prefixes (Table 0.A.4) are also

equally applied to the cgs system (note that the base unit, cm, already has a prefix).

Table 0.A.1. Selected cgs – SI conversionsa

Dimension cgs unit (abbrev.) Factor SI unitb (abbrev.)

Length centimetre (cm) 102 metre (m)

Mass gram (g) 103 kilogram (kg)

Time second (s) 1 second (s)

Energy erg (erg) 107 joule (J)

Power erg second1 (erg s1) 107 watt (W)

Temperature kelvin (K) 1 kelvin (K)

Force dyne (dyn) 105 newton (N)

Pressure dyne centimetre2 0.1 newton metre2 (N m2)

(dyn cm2)

barye ðbaÞc 0.1 pascal (Pa)

Magnetic flux density (field)d gauss (G) 104 tesla (T)

Angle radian (rad) 1 radian (rad)

Solid angle steradian (sr) 1 steradian (sr)

a Value in cgs units times factor equals value in SI units.b Systeme International d’Unites.c This unit is rarely used in astronomy in favour of dyn cm2.d See notes to Table I.1.

xxii INTRODUCTION

Table 0.A.2. Examples of equivalent units

Equationa Name Units

P ¼ n k T Ideal Gas Law dyn cm2 ¼ 1

cm3

erg

K

Kð Þ

¼ erg cm3

F ¼ m a Newton’s Second Law dyn ¼ g cm s2

W ¼ F s Work Equation erg ¼ dyn cm ¼ g cm2 s2

Ek ¼ 12

m v2 Kinetic Energy Equation erg ¼ g cm2 s2

EPG ¼ m g h Gravitational Potential Energy Equation erg ¼ g cm2 s2

EPe ¼q1 q2

rElectrostatic Potential Energy Equation erg ¼ esu2 cm1

See Table 0.A.3 for the meaning of the symbols and Table G.2 for the meaning of the constants.

Table 0.A.3. List of symbols

Symbol Meaning

a radius, acceleration

A atomic weight

A area, albedo, total extinction

B magnetic flux density, magnetic field strengtha

BðTÞ intensity of a black body (or specific intensity if subscripted with n or l)

Dp degree of polarization

e charge of the electron

E energy, selective extinction

E electric field strength

EM emission measure

fi;j oscillator strength between levels, i and j

f correction factor

F; S; f flux (or flux density, if subscripted with n or l)

F force

gn statistical weight of level, n

g Gaunt factor

I intensity (or specific intensity, if subscripted with n or l)

jn emission coefficient

J mean intensity (or mean specific intensity, if subscripted with n or l)

JðEÞ cosmic ray flux density per unit solid angle

l mean free path

L luminosity (or spectral luminosity, if subscripted with n or l)

m, M apparent magnitude & absolute magnitude, respectively

M; m mass

n index of refraction, principal quantum number

n, N number density and number of (object), respectively

N map noise

N number of moles, column density

p momentum, electric dipole moment

P power (or spectral power, if subscripted with n or l), probability

P pressure

q charge

(Continued)

APPENDIX xxiii

Almost all equations used in this text look identical in the two systems and one need

only ensure that the constants and input parameters are all consistently used in the

adopted system. There are, however, a few cases in which the equation itself changes

between cgs and SI. An example is the Coulomb (electrostatic) force,

F ¼ q1 q2

r2ð0:A:1Þ

Table 0.A.3. (Continued)

Symbol Meaning

Q efficiency factor

r, d, D, s, x, y, z, l, h, R Position, separation, or distance

R Rydberg constant

R collision rate

S source function

t time

T kinetic energy

T temperature

T period

u energy density (or spectral energy density, if subscripted with n or l)

U, V, B, etc. apparent magnitudes

U excitation parameter

v velocity, speed

V volume

X; Y; Z mass fraction of hydrogen, helium and heavier elements, respectively

z redshift, zenith angle

Z atomic number

a synchrotron spectral index, fine structure constant

an absorption coefficient

ar recombination coefficient

g Lorentz factor

gcoll collision rate coefficient

G spectral index of cosmic ray power law dist’n, damping constant

e permittivity, cosmological energy density

u, f one-dimensional angle

kn mass absorption coefficient

l wavelength

m permeability, mean molecular weight

n frequency, collision rate per unit volume

r mass density

s cross-sectional area, Stefan-Boltzmann constant, Gaussian dispersion

tn, t optical depth & timescale, respectively

F line shape function

x ionization potential

v angular frequency

V solid angle, cosmological mass or energy density

a See notes to Table I.1.

xxiv INTRODUCTION

With the two charges, q1 and q2 expressed in electrostatic units (esu, see Table G.2),

and the separation, r, in cm, the answer will be in dynes. Note that there is no constant

of proportionality in this equation, unlike the SI equivalent (Prob. 0.1). Equations in the

cgs system show the most difference with their SI equivalents when electric and

magnetic quantities are used. For example, in the cgs system, the permittivity and

permeability of free space, e0 and m0, respectively, are both unitless and equal to 1 (see

also notes to Table I.1).

Astronomers also have a number of units specific to the discipline. There are a

variety of reasons for this, but the most common involves a process of normalisation.

The value of some parameter is expressed in comparison to another known, or at least

more familiar value. Some examples (see Table G.3) are the astronomical unit (AU)

which is the distance between the Earth and the Sun. It is much easier to visualise the

distance to Pluto as 40 AU than as 6 1014 cm. Expressing the masses of stars and

galaxies in Solar masses (M) or an object’s luminosity in Solar luminosities (L) is

also very common. When such units are used, the parameter is often written as M=M,

or L=L. Some examples can be seen in Eqs. 8.15 through 8.18 and in many other

equations in this book.

A very valuable tool for checking the answer to a problem, or to help understand

an equation, is that of dimensional analysis. The dimensions of an equation (e.g.

time, velocity, distance) must agree and therefore their units (s, cm s1, cm,

respectively) must also agree. Two quantities can be added or subtracted only if

they have the same units, and logarithms and exponentials are unitless. In this

process, it is helpful to recall some equivalent units which are revealed by writing

Table 0.A.4. SI prefixes

Name Prefix Factor

yocto y 1024

zepto z 1021

atto a 1018

femto f 1015

pico p 1012

nano n 109

micro m 106

milli m 103

centi c 102

deci d 101

kilo k 103

mega M 106

giga G 109

tera T 1012

peta P 1015

exa E 1018

zeta Z 1021

yota Y 1024

APPENDIX xxv

down some simple well-known equations in physics. A few examples are provided

in Table 0.A.2. The example of the Ideal Gas Law in this table also shows the

process of dimensional analysis, which involves writing down the units to every

term and then cancelling where possible. A more complex example of dimensional

analysis is given in Example A.1.

Example 0.A.1

For a gas in thermal equilibrium at some uniform temperature, T, and uniform density, n,

the number density of particles with speeds2 between v and v þ dv is given by the

Maxwell–Boltzmann (or simply ‘Maxwellian’) velocity distribution,

nðvÞ dv ¼ nm

2p k T

3=2

exp m v2

2 k T

4p v2 dv ð0:A:2Þ

where nðvÞ is the gas density per unit velocity interval, m is the mass of a gas particle (taken

here to be the same for all particles), and v is the particle speed. A check of the units gives,

1

cm3 cms

cm

s¼ 1

cm3

gergK

K

3=2

exp g ðcm

sÞ2

ergK

K

!cm

s

2 cm

sð0:A:3Þ

Simplifying yields,

1

cm3¼ 1

s3

g

erg

3=2

exp g ðcm

sÞ2

erg

!ð0:A:4Þ

Using the equivalent units for energy (Table A.2), the exponential is unitless, as required,

and we find,

1

cm3¼ 1

cm3ð0:A:5Þ

Figure 3.12 shows a plot of this function. If Eq. (0.A.2) is integrated over all velocities,

either on the left hand side (LHS) or the right hand side (RHS), the total density should

result. Since an integration over all velocities is equivalent to a sum over the individual

infinitesimal velocity intervals, the total density will also have the required units of 1cm3.

This dimensional analysis helps to clarify the fact that, since the total density, n, appears on

the RHS of Eq. (0.A.2) and is a constant, an integration over all terms, except n, on the RHS

must be unitless and equal 1. (In fact these remaining terms represent a probability

distribution function, see Sect. 3.4.1 for a description.) Also, since the number density is

just the number of particles, N, divided by a constant (the volume), we could have

substituted NðvÞ dv on the LHS and N on the RHS for the density terms. Similarly, since

2Speed and velocity are taken to be equivalent in this text unless otherwise indicated.

xxvi INTRODUCTION

the density and temperature are constant, we could have multiplied Eq. (0.A.2) by k T to

turn it into an equation for the particle pressure in a velocity interval PðvÞ dv with the total

pressure, P on the RHS (Table 0.A.2). Thus, while dimensional analysis says nothing about

the origin or fundamentals of an equation, it can go a long way in revealing the meaning of

one and how it might be manipulated.

An important comment is that one should be very careful of simplifying units

without thinking about their meaning. A good example is the unit, erg s1 Hz1 which

is a representation of a luminosity or power (see Sect. 1.1) per unit frequency (Hz) in

some waveband. Since Hz can be represented as s1, the above could be written

erg s1 s1 ¼ erg which is simply an energy and does not really express the intended

meaning of the term. Similar difficulties can arise when a term is expressed as ‘per cm

of waveband’. Units of frequency or wavelength should not be combined with units of

time or distance, respectively.

APPENDIX xxvii

PART IThe Signal Observed

The radiation from an astronomical source can be thought of as a signal that provides us

with information about it. In order to relate the signal received at the Earth to the

physical conditions within an astronomical source, however, we first need ways to

describe and measure light. This requires setting out the basic definitions for quantities

involving light and the relationships between them. The definitions range from those

associated with values measured at the Earth to those that are intrinsic to the source

itself. We do this without regard (yet) for the processes that actually generate the light,

an approach similar to what is often followed in Mechanics. For example, first one

studies Kinematics which relates distances, velocities, and accelerations and the

relations between them. Later, one considers Dynamics which deals with the forces

that produce these motions. Measuring light involves a deep understanding of how the

measurement process itself affects the signal and also how the Earth’s atmosphere

interferes. Our instrumentation imposes its own signature on an astronomical signal

and it is important to account for this imposition. These steps are fundamental and lay

the groundwork for turning the measurement of a weak glimmer of light into an

understanding of what drives the most powerful objects in the Universe.

1Defining the Signal

. . . the distance of the invisible background [is] so immense that no ray from it has yet been able to

reach us at all.

–Edgar Allan Poe in Eureka, 1848

1.1 The power of light – luminosity and spectral power

The luminosity, L, of an object is the rate at which the object radiates away its energy

(cgs units of erg s1 or SI units of watts),

dE ¼ L dt ð1:1Þ

This quantity has the same units as power and is simply the radiative power output from

the object. It is an intrinsic quantity for a given object and does not depend on the

observer’s distance or viewing angle. If a star’s luminosity is L at its surface, then at a

distance r away, its luminosity is still L.Any object that radiates, be it spherical or irregularly shaped, can be described by

its luminosity. The Sun, for example, has a luminosity of L ¼ 3:85 1033 erg s1

(Table G.3), most of which is lost to space and not intercepted by the Earth

(Example 1.1).

Example 1.1

Determine the fraction of the Sun’s luminosity that is intercepted by the Earth. Whatluminosity does this correspond to?

At the distance of the Earth, the Sun’s luminosity, L, is passing through the imaginary

surface of a sphere of radius, r ¼ 1 AU. The Earth will be intercepting photons over only

Astrophysics: Decoding the Cosmos Judith A. Irwin# 2007 John Wiley & Sons, Inc. ISBN: 978-0-470-01305-2(HB) 978-0-470-01306-9(PB)

the cross-sectional area that is facing the Sun. This is because the Sun is so far away that

incoming light rays are parallel. Thus, the fraction will be

f ¼ R2

4 r2

ð1:2Þ

where R is the radius of the Earth. Using the values of Table G.3, the fraction is

f ¼ 4:5 1010 and the intercepted luminosity is therefore Lint ¼ f L ¼ 1:731024 erg s1. A hypothetical shell around a star that would allow a civilization to intercept

all of its luminosity is called a Dyson Sphere (Figure 1.1).

When one refers to the luminosity of an object, it is the bolometric luminosity that is

understood, i.e. the luminosity over all wavebands. However, it is not possible to

determine this quantity easily since observations at different wavelengths require

different techniques, different kinds of telescopes and, in some wavebands, the

Sun

Mercury

Venus

Rad

ius;

1.5

x 1

08 km

Dyson Sphere

Infrared Radiation

Figure 1.1. Illustration of a Dyson Sphere that could capture the entire luminous ouput from theSun. Some have suggested that advanced civilizations, if they exist, would have discovered ways tobuild such spheres to harness all of the energy of their parent stars.

4 CH1 DEFINING THE SIGNAL

necessity of making measurements above the obscuring atmosphere of the Earth. Thus,

it is common to specify the luminosity of an object for a given waveband (see

Table G.6). For example, the supernova remnant, Cas A (Figure 1.2), has a radio

luminosity (from 1 ¼ 2 107 Hz to 2 ¼ 2 1010 Hz) of Lradio ¼ 3 1035 erg s1

(Ref. [6]) and an X-ray luminosity (from 0:3 to 10 keV) of LX-ray ¼ 3 1037 erg s1

(Ref. [37]). Its bolometric luminosity is the sum of these values plus the luminosities

from all other bands over which it emits. It can be seen that the radio luminosity

might justifiably be neglected when computing the total power output of Cas A.

Clearly, the source spectrum (the emission as a function of wavelength) is of

some importance in understanding which wavebands, and which processes, are most

important in terms of energy output. The spectrum may be represented mostly by

continuum emission as implied here for Cas A (that is, emission that is continuous

over some spectral region), or may include spectral lines (emission at discrete

wavelengths, see Chapter 3, 5, or 9). Even very weak lines and weak continuum

emission, however, can provide important clues about the processes that are occurring

within an astronomical object, and must not be neglected if a full understanding of the

source is to be achieved.

In the optical region of the spectrum, various passbands have been defined

(Figure 1.3). The Sun’s luminosity in V-band, for example, represents 93 per cent of

its bolometric luminosity.

The spectral luminosity or spectral power is the luminosity per unit bandwidth and

can be specified per unit wavelength, L (cgs units of erg s1 cm1) or per unit

Figure 1.2. The supernova remnant, Cas A, at a distance of 3:4 kpc and with a linear diameter of4 pc, was produced when a massive star exploded in the year AD 1680. It is currently expandingat a rate of 4000 km s1 (Ref. [181]) and the proper motion (angular motion in the plane of the sky,see Sect. 7.2.1.1) of individual filaments have been observed. One side of the bipolar jet, emanatingfrom the central object, can be seen at approximately 10 o’clock. (a) Radio image at 20 cm shownin false colour (see Sect. 2.6) from Ref. [7]. Image courtesy of NRAO/AUI/NSF. (b) X-ray emission,with red, green and blue colours showing, respectively, the intensity of low, medium and highenergy X-ray emission. (Reproduced courtesy of NASA/CXC/SAO) (see colour plate)

1.1 THE POWER OF LIGHT – LUMINOSITY AND SPECTRAL POWER 5

frequency, L (erg s1 Hz1),

dL ¼ L d ¼ L d ð1:3Þ

so L ¼Z

L d ¼Z

L d ð1:4Þ

Note that, since ¼ c ;

d ¼ c

2d ð1:5Þ

so the magnitudes of L and L will not be equal (Prob. 1.1). The negative sign in

Eq. (1.5) serves to indicate that, as wavelength increases, frequency decreases.

In equations like Eq. (1.4) in which the wavelength and frequency versions of a function

are related to each other, this negative is already taken into account by ensuring that the

lower limit to the integral is always the lower wavelength or frequency. Note that the cgs

units of L (erg s1 cm1) are rarely used since 1 cm of bandwidth is exceedingly large

(Table G.6). Non-cgs units, such as erg s1 A1 are sometimes used instead.

0

0.2

0.4

0.6

0.8

1

300 400 500 600 700 800 900

(a)

U B V R I

Filt

er R

esp

on

se

Wavelength (nm)

0

0.2

0.4

0.6

0.8

1

1000 1500 2000 2500 3000 3500 4000

(b)

J H K L L*

Wavelength (nm)

Figure 1.3. Filter bandpass responses for (a) the UBVRI bands (Ref. [17]) and (b) the JHKLL

bands (Ref. [19]). (The U and B bands correspond to UX and BX of Ref. [17].) Corresponding datacan be found in Table 1.1


Luminosity is a very important quantity because it is a basic parameter of the source

and is directly related to energetics. Integrated over time, it provides a measure of the

energy required to make the object shine over that timescale. However, it is not a

quantity that can be measured directly and must instead be derived from other

measurable quantities that will shortly be described.

1.2 Light through a surface – flux and flux density

The flux of a source, f (erg s1 cm2), is the radiative energy per unit time passing

through unit area,

dL ¼ f dA ð1:6Þ

As with luminosity, we can define a flux in a given waveband or we can define it per unit

spectral bandwidth. For example, the spectral flux density, or just flux density

(erg s1 cm2 Hz1 or erg s1 cm2 cm1)1 is the flux per unit spectral bandwidth,

either frequency or wavelength, respectively,

dL ¼ f dA dL ¼ f dA

df ¼ f d df ¼ f d ð1:7Þ

A special unit for flux density, called the Jansky (Jy) is utilized in astronomy, most

often in the infrared and radio parts of the spectrum,

1 Jy ¼ 1026 W m2 Hz1 ¼ 1023 erg s1 cm2 Hz1 ð1:8Þ

Radio sources that are greater than 1 Jy are considered to be strong sources by

astronomical standards (Prob. 1.3).

The spectral response is independent of other quantities such as area or time so

Eq. (1.6) and the first line of Eq. (1.7) show the same relationships except for the

subscripts. To avoid repitition, then, we will now give the relationships for the

bolometric quantities and it will be understood that these relationships apply to the

subscripted ‘per unit bandwidth’ quantities as well.

The luminosity, L, of a source can be found from its flux via,

L ¼Z

f dA ¼ 4 r2 f ð1:9Þ

where r is the distance from the centre of the source to the position at which the flux

has been determined. The 4 r2 on the right hand side (RHS) of Eq. (1.9) is strictly

1The two ‘cm’ designations should remain separate. See the Appendix at the end of the Introduction.

1.2 LIGHT THROUGH A SURFACE – FLUX AND FLUX DENSITY 7

only true for sources in which the photons that are generated can escape in all directions,

or isotropically. This is usually assumed to be true, even if the source itself is irregular in

shape (Figure 1.4). These photons pass through the imaginary surfaces of spheres as they

travel outwards. The 1r2 fall-off of flux is just due to the geometry of a sphere (Figure

1.5.a). In principle, however, one could imagine other geometries. For example, the flux

of a man-made laser beam would be constant with r if all emitted light rays are parallel

and without losses (Figure 1.5.b). Light that is beamed into a narrow cone, such as may

be occurring in pulsars2 is an example of an intermediate case (Prob. 1.4).

For stars, we now define the astrophysical flux, F, to be the flux at the surface of the

star,

L ¼ 4R2 F ¼ 4 r2 f ) f ¼ R

r

2

F ð1:10Þ

where L is the star’s luminosity and R is its radius.

Using values from Table G.3, astrophysical flux of the Sun is F ¼ 6:331010 erg s1 cm2 and the Solar Constant, which is the flux of the Sun at the distance

Figure 1.4. An image of the Centaurus A jet emanating from an active galactic nucleus (AGN) atthe centre of this galaxy and at lower right of this image. Radio emission is shown in red and X-raysin blue. (Reproduced by permission of Hardcastle M.J., et al., 2003 ApJ, 593, 169.) Even thoughgaseous material may be moving along the jet in a highly directional fashion, the RHS of Eq. (1.9)may still be used, provided that photons generated within the jet (such as at the knot shown)escape in all directions. (see colour plate)

2Pulsars are rapidly spinning neutron stars with strong magnetic fields that emit their radiation in beamed cones.Neutron stars typically have about the mass of the Sun in a diameter only tens of km across.


of the Earth3, denoted, S, is 1:367 106 erg s1 cm2. The Solar Constant is of great

importance since it is this flux that governs climate and life on Earth. Modern satellite

data reveal that the solar ‘constant’ actually varies in magnitude, showing that our Sun

is a variable star (Figure 1.6). Earth-bound measurements failed to detect this variation

since it is quite small and corrections for the atmosphere and other effects are large in

comparison (e.g. Prob. 1.5).

The flux of a source in a given waveband is a quantity that is measurable, provided

corrections are made for atmospheric and telescopic responses, as required (see Sects.

2.2, 2.3). If the distance to the source is known, its luminosity can then be calculated

from Eq. (1.9).

1.3 The brightness of light – intensity and specific intensity

The intensity, I (erg s1 cm2 sr1), is the radiative energy per unit time per unit solid

angle passing through a unit area that is perpendicular to the direction of the emission.

The specific intensity (erg s1 cm2 Hz1 sr1 or erg s1 cm2 cm1 sr1) is the

radiative energy per unit time per unit solid angle per unit spectral bandwidth (either

frequency or wavelength, respectively) passing through unit area perpendicular to the

direction of the emission. The intensity is related to the flux via,

df ¼ I cos d ð1:11Þ

Laser

Source

r1

r2

(a)

(b)

Figure 1.5. (a) Geometry illustrating the 1r2 fall-off of flux with distance, r, from the source. The

two spheres shown are imaginary surfaces. The same amount of energy per unit time is goingthrough the two surface areas shown. Since the area at r2 is greater than the one at r1, the energyper unit time per unit cm2 is smaller at r2 than r1. Since measurements are made over size scales somuch smaller than astronomical distances, the detector need not be curved. (b) Geometry of an

artificial laser. For a beam with no divergence, the flux does not change with distance.

3This is taken to be above the Earth’s atmosphere.

1.3 THE BRIGHTNESS OF LIGHT – INTENSITY AND SPECIFIC INTENSITY 9

As before, the same kind of relation could be written between the quantities per unit

bandwidth, i.e. between the specific intensity and the flux density.

The specific intensity, I , is the most basic of radiative quantities. Its formal

definition is written,

dE ¼ I cos d d dA dt ð1:12Þ

Note that each elemental quantity is independent of the others so, when integrating, it

doesn’t matter in which order the integration is done.

The intensity isolates the emission that is within a given solid angle and at some

angle from the perpendicular. The geometry is shown in Figure 1.7 for a situation in

which a detector is receiving emission from a source in the sky and for a situation in

which an imaginary detector is placed on the surface of a star. In the first case, the

source subtends some solid angle in the sky in a direction, , from the zenith. The

factor, cos accounts for the foreshortening of the detector area as emission falls on it.

Figure 1.6. Plot of the Solar Constant (in W m2) as a function of time from satellite data. Thevariation follows the 11-year Sunspot cycle such that when there are more sunspots, the Sun, onaverage, is brighter. The peak to peak variation is less than 0.1 per cent. This plot providesdefinitive evidence that our Sun is a variable star. (Reproduced by permission of www.answers.com/topic/solar-variation)


Usually, a detector would be pointed directly at the source of interest in which case

cos ¼ 1. In the second case, the coordinate system has been placed at the surface

of a star. At any position on the star’s surface, radiation is emitted over all directions

away from the surface. The intensity refers to the emission in the direction,

radiating into solid angle, d. Example 1.2 indicates how the intensity relates

to the flux for these two examples. Figure 1.7 also helps to illustrate the generality of

these quantities. One could place the coordinate system at the centre of a star, in

interstellar space, or wherever we wish to determine these radiative properties of a

source (Probs. 1.6, 1.7).

Example 1.2

(a) A detector pointed directly at a uniform intensity source in the sky of small solid angle,

, would measure a flux,

f ¼Z

I cos d I ð1:13Þ

(b) The astrophysical flux at the surface of an object (e.g. a star) whose radiation is

escaping freely at all angles outwards (i.e. over 2 sr), can be calculated by integrating

dA

dA cosθ

d Ω (a) (b)

θ

θ

dA

dA cosθ

d Ω

θ

θ

Figure 1.7. Diagrams showing intensity and its dependence on direction and solid angle,using a spherical coordinate system such as described in Appendix B. (a) Here dA would be anelement of area of a detector on the Earth, the perpendicular upwards direction is towards thezenith, a source is in the sky in the direction, , and d is an elemental solid angle on thesource. The arrows show incoming rays from the centre of the source that flood the detector. (b) In

this example, an imaginary detector is placed at the surface of a star. At each point on the

surface, photons leave in all directions away from the surface. The intensity would be a

measure of only those photons which pass through a given solid angle at a given angle, fromthe vertical

1.3 THE BRIGHTNESS OF LIGHT – INTENSITY AND SPECIFIC INTENSITY 11

in spherical coordinates (see Appendix B),

F ¼Z

I cos d ¼Z 2

0

Z 2

0

I cos sin d d ¼ I ð1:14Þ

Figure 1.8 shows a practical example as to how one might calculate the flux of a

source for a case corresponding to Example 1.2a, but for which the intensity varies with

position. The intensity in a given waveband is a measurable quantity, provided a solid

angle can also be measured. If a source is so small or so far away that its angular size

cannot be discerned (i.e. it is unresolved, see Sects. 2.2.3, 2.2.4, 2.3.2), then the

intensity cannot be determined. In such cases, it is the flux that is measured, as shown

in Figure 1.9. All stars other than the Sun would fall into this category4.

Specific intensity is also referred to as brightness which has its intuitive meaning. A

faint source has a lower value of specific intensity than a bright source. Note that it is

possible for a source that is faint to have a larger flux density than a source that is bright

if it subtends a larger solid angle in the sky (Prob. 1.9).

Figure 1.8. Looking directly at a hypothetical object in the sky corresponding to the situationshown in Figure 1.7.a but for ¼ 0 (i.e. the detector pointing directly at the source). The objectsubtends a total solid angle, , which is small and therefore 0 at any location on the source.In this example, the object is of non-uniform brightness and is split up into many small squaresolid angles, each of size, i and within which the intensity is Ii. Then we can approximatef ¼

RI cos d using f

PIi i. Basically, to find the flux, we add up the individual fluxes of

all elements.

4The exception is a few nearby stars for which special observing techniques are required.


The intensity and specific intensity are independent of distance (constant with

distance) in the absence of any intervening matter5. The easiest way to see this is

via Eq. (1.13). Both f and decline as 1r2 (Eq. (1.11), Eq. (B.2), respectively) and

therefore I is constant with distance. The Sun, for example, has I ¼ F= ¼2:01 1010 erg s1 cm2 sr1 as viewed from any source at which the Sun subtends

a small, measurable solid angle. The constancy of I with distance is general, however,

applying to large angles as well. This is a very important result, since a measurement of

I allows the determination of some properties of the source without having to know its

distance (e.g. Sect. 4.1).

1.4 Light from all angles – energy density and mean intensity

The energy density, u (erg cm3), is the radiative energy per unit volume. It describes

the energy content of radiation in a unit volume of space,

du ¼ dE

dVð1:15Þ

The specific energy density is the energy density per unit bandwidth and, as usual,

u ¼Ru d ¼

Ru d. The energy density is related to the intensity (see Figure 1.10,

Eq. 1.12) by,

u ¼ 1

c

ZI d ¼ 4

cJ ð1:16Þ

Figure 1.9. In this case, a star has a very small angular size (left) and so, when detected in asquare solid angle, p (right), which is determined by the properties of the detector, its light is‘smeared out’ to fill that solid angle. In such a case, it is impossible to determine the intensity ofthe surface of the star. However, the flux of the star, f, is preserved, i.e. f ¼ I ¼ I p

(Eq. 1.13) where I is the true intensity of the star, is the true solid angle subtended by the star,and I is the mean intensity in the square. Thus, for an object of angular size smaller than can beresolved by the available instruments (see Sects. 2.2.3, 2.2.4, and 2.3.2), we measure the flux(or flux density), but not intensity (or specific intensity) of the object

5More accurately, I=n2 is independent of distance along a ray path, where n is the index of refraction but thedifference is negligible for our purposes.

1.4 LIGHT FROM ALL ANGLES – ENERGY DENSITY AND MEAN INTENSITY 13

where J is the mean intensity, defined by,

J 1

4

ZI d ð1:17Þ

The mean intensity is therefore the intensity averaged over all directions. In an

isotropic radiation field, J ¼ I. In reality, radiation fields are generally not isotropic,

but some are close to it or can be approximated as isotropic, for example, in the centres

of stars or when considering the 2.7 K cosmic microwave background radiation (Sect.

3.1). In a non-isotropic radiation field, J is not constant with distance, even though I is.

Example 1.3 provides a sample computation.

Example 1.3

Compute the mean intensity and the energy density at the distance of Mars. Assume that theonly important source is the Sun.

J ¼ 1

4

Z 4

0

I d

¼ 1

4

Z

I d I

4¼ I

4

2

4¼ I

16

2R

rMars

2

ð1:18Þ

where we have used Eq. (B.3) to express the solid angle in terms of the linear angle, and

Eq. (B.1) to express the linear angle in terms of the size of the Sun and the distance of Mars.

Inserting I ¼ 2:01 1010 erg s1 cm2 sr1 (Sect. 1.3), R ¼ 6:96 1010 cm, and

rMars ¼ 2:28 1013 cm (Tables G.3, G.4), we find, J ¼ 4:7 104 erg s1 cm2 sr1.

Then u ¼ 4cð4:7 104Þ ¼ 2:0 105 erg cm3.

The radiation field (u or J) in interstellar space due to randomly distributed stars

(Prob. 1.10) must be computed over a solid angle of 4 steradians, given that

θ θ

dA

Idl

dV = dAdl

dl = cdt

cos θ

Figure 1.10. This diagram is helpful in relating the energy density (the radiative energy per unitvolume) to the light intensity. An individual ray spends a time, dt ¼ dl=ðc cos Þ in an infinitesimalcylindrical volume of size, dV ¼ dl dA. Combined with Eq. (1.12), the result is Eq. (1.16).


starlight contributes from many directions in the sky. However, in this case, J 6¼ I

because there is no emission from directions between the stars. If there were so

many stars that every line of sight eventually intersected the surface of a star of

brightness, I, then J ¼ I and the entire sky would appear as bright as I. This

would be true even if the stars were at great distances since I, being an intensity, is

independent of distance. If this is the case, we would say that the stellar covering

factor is unity.

A variant of this concept is called Olbers’ Paradox after the German astron-

omer, Heinrich Wilhelm Olbers who popularized it in the 19th century. It was

discussed as early as 1610, though, by the German astronomer, Johannes Kepler,

and was based on the idea of an infinite starry Universe which had been propounded

by the English astronomer and mathematician, Thomas Digges, around 1576. If the

Universe is infinite and populated throughout with stars, then every line of sight

should eventually intersect a star and the night sky as seen from Earth should be

as bright as a typical stellar surface. Why, then, is the night sky dark?6 Kepler took

the simple observation of a dark night sky as an argument for the finite extent of the

Universe, or at least of its stars. The modern explanation, however, lies with the

Figure 1.11. Why is the night sky dark? If the Universe is infinite and populated in all directionsby stars, then eventually every sight line should intersect the surface of a star. Since I is constantwith distance, the night sky should be as bright as the surface of a typical star. This is known asOlbers’ Paradox, though Olbers was not the first to note this discrepancy. See Sect. 1.4.

6The earlier form of the question was posed somewhat differently, referring to increasing numbers of stars onincreasingly larger shells with distance from the Earth.

1.4 LIGHT FROM ALL ANGLES – ENERGY DENSITY AND MEAN INTENSITY 15

intimate relation between time and space on cosmological scales (Sect. 7.1). Since

the speed of light is constant, as we look farther into space, we also look farther back

in time. The Universe, though, is not infinitely old but rather had a beginning

(Sect. 3.1) and the formation of stars occurred afterwards. The required number of

stars for a bright night sky is 1060 and the volume needed to contain this quantity of

stars implies a distance of 1023 light years (Ref. [74]). This means that we need to see

stars at an epoch corresponding to 1023 years ago for the night sky to be bright. The

Universe, however, is younger than this by 13 orders of magnitude (Sect. 3.1)! Thus,

as we look out into space and back in time, our sight lines eventually reach an epoch

prior to the formation of the first stars when the covering factor is still much less than

unity. (Today, we refer to this epoch as the dark ages.) Remarkably, this solution was

hinted at by Edgar Allan Poe in his prose-poem, Eureka in 1848 (see the prologue to

this chapter).

1.5. How light pushes – radiation pressure

Radiation pressure is the momentum flux of radiation (the rate of momentum transfer

due to photons, per unit area). It can also be thought of as the force per unit area exerted

by radiation and, since force is a vector, we will treat radiation pressure in this way

as well7. Thus, the pressure can be separated into its normal, P?, and tangential,

Pk, components with respect to the surface of a wall. The normal radiation pressure will

be,

dP? ¼ dF?dA

¼ dp

dt dAcos ¼ dE

c dt dAcos ð1:19Þ

where we have expressed the momentum of a photon in terms of its energy (Table I.1).

Using Eq. (1.12) we obtain,

dP? ¼ 1

c

I cos2 d ð1:20Þ

For the tangential pressure, we use the same development but take the sine of the

incident angle, yielding,

dPk ¼1

c

I cos sin d ð1:21Þ

7Pressure is actually a tensor which is a mathematical quantity described by a matrix (a vector is a specific kind oftensor). We do not need a full mathematical treatment of pressure as a tensor, however, to appreciate the meaningof radiation pressure.


Then for isotropic radiation,

P? ¼ 1

c

Z4

I cos2 d ¼ 4

3 cI

Pk ¼1

c

Z4

I cos sin d ¼ 0

therefore P ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP?

2 þ Pk2

q¼ 4

3 cI ¼ 1

3u ð1:22Þ

where we have used a spherical coordinate system for the integration (Appendix B),

Eq. (1.16), and the fact that J ¼ I in an isotropic radiation field. Note that the units of

pressure are equivalent to the units of energy density, as indicated in Table 0.A.2. Since

photons carry momentum, the pressure is not zero in an isotropic radiation field. A

surface placed within an isotropic radiation field will not experience a net force,

however. This is similar to the pressure of particles in a thermal gas. There is no net

force in one direction or another, but there is still a pressure associated with such a gas

(Sect. 3.4.2).

We can also consider a case in which the incoming radiation is from a fixed angle,

and the solid angle subtended by the radiation source,, is small. This would result in

an acceleration of the wall, but the result depends on the kind of surface the photons are

hitting. We consider two cases, illustrated in Figure 1.12: that in which the photon loses

all of its energy to the wall (perfect absorption) and that in which the photon loses none

of its energy to the wall (perfect reflection).

For perfect absorption, integrating Eqs. (1.20), (1.21) with , , constant, yields,

P? ¼ 1

c

I cos2 ¼ f

ccos2

Pk ¼1

c

I cos sin ¼ f

ccos sin

P ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP?

2 þ Pk2

q¼ f

ccos ð1:23Þ

where we have used Eq. (1.13) with f the flux along the directed beam8.

For perfect reflection, only the normal component will have any effect against the

wall (as if the surface were hit by a ball that bounces off). Also, because the momentum

of the photon reverses direction upon reflection, the change in momentum is twice the

value of the absorption case. Thus, the situation can be described by Eq. (1.20) except

8For a narrow beam, this is equivalent to the Poynting flux (Table I.1).

1.5. HOW LIGHT PUSHES – RADIATION PRESSURE 17

for a factor of 2.

P ¼ P? ¼ 2

c

I cos2 ¼ 2 f

ccos2 ð1:24Þ

A comparison of Eqs. (1.23) and (1.24) shows that, provided the incident angle is not

very large, a reflecting surface will experience a considerably greater radiation force

than an absorbing surface. Moreover, as Figure 1.12 illustrates, the direction of the

surface is not directly away from the source of radiation as it must be for the absorbing

case. These principles are fundamental to the concept of a Solar sail (Figure 1.13).

Figure 1.13. Artist’s conception of a thin, square, reflective Solar sail, half a kilometre across.Reproduced by permission of NASA/MSFC

reflected ray

incident ra

y

area A

Freflection θ

F absorptio

n

Figure 1.12. An incoming photon exerts a pressure on a surface area. For perfect absorption, thearea will experience a force in the direction, ~Fabsorption. For perfect reflection, only the normalcomponent of the force is effective and the resulting force will be in the direction, ~F reflection


Since the direction of motion depends on the angle between the radiation source and

the normal to the surface, it would be possible to ‘tack’ a Solar sail by altering the angle

of the sail, in a fashion similar to the way in which a sailboat tacks in the wind.

Moreover, even if the acceleration is initially very small, it is continuous and thus very

high velocities could eventually be achieved for spacecraft designed with Solar sails

(Prob. 1.11).

1.6 The human perception of light – magnitudes

Magnitudes are used to characterize light in the optical part of the spectrum, including

the near IR and near UV. This is a logarithmic system for light, similar to decibels for

sound, and is based on the fact that the response of the eye is logarithmic. It was first

introduced in a rudimentary form by Hipparchus of Nicaea in about 150 B.C. who

labelled the brightest stars he could see by eye as ‘first magnitude’, the second brightest

as ‘second magnitude’, and so on. Thus began a system in which brighter stars have

lower numerical magnitudes, a sometimes confusing fact. As the human eye has been

the dominant astronomical detector throughout most of history, a logarithmic system

has been quite appropriate. Today, the need for such a system is less obvious since the

detector of choice is the CCD (charge coupled device, Sect. 2.2.2) whose response is

linear. However, since magnitudes are entrenched in the astronomical literature, still

widely used today, and well-characterized and calibrated, it is very important to

understand this system.

1.6.1 Apparent magnitude

The apparent magnitude and its corresponding flux density are values as measured

above the Earth’s atmosphere or, equivalently, as measured from the Earth’s surface,

corrected for the effects of the atmosphere,

m m0¼ 2:5 log

f

f0

m m0¼ 2:5 log

f

f0

ð1:25Þ

where the subscript, 0, refers to a standard calibrator used as a reference, m and m are

apparent magnitudes in some waveband and f, f are flux densities in the same band.

Note that this equation could also be written as a ratio of fluxes, since this would only

require multiplying the flux density (numerator and denominator) by an effective

bandwidth to make this conversion. This system is a relative one, such that the

magnitude of the star of interest can be related to that of any other star in the same

waveband via Eq. (1.25). For example, if a star has a flux density that is 100 times greater

1.6 THE HUMAN PERCEPTION OF LIGHT – MAGNITUDES 19

than a second star, then its magnitude will be 5 less than the second star. However, in

order to assign a specific magnitude to a specific star, it is necessary to identify certain

standard stars with known flux densities to which all others can be compared.

Several slightly different calibration systems have evolved over the years so, for

careful and precise work, it is necessary to specify which system is being used when

measuring or stating a magnitude. An example of such a system is the UBVRIJHKL

Cousins–Glass–Johnson system for which parameters are provided in Table 1.1.

The corresponding wavebands, U, B, V, etc., are illustrated in Figure 1.3. The apparent

magnitude is commonly written in such a way as to specify these wavebands

directly, e.g.

V V0 ¼ 2:5 logfV

fV0

B B0 ¼ 2:5 logfB

fB0

etc: ð1:26Þ

where the flux densities can be expressed in either their -dependent or -dependent

forms. The V-band, especially, since it corresponds to the waveband in which the eye is

most sensitive (cf. Table G.5), has been widely and extensively used. Some examples of

apparent magnitudes are provided in Table 1.2.

The standard calibrator in most systems has historically been the star, Vega. Thus,

Vega would have a magnitude of 0 in all wavebands (i.e. U0 ¼ 0, B0 ¼ 0, etc) and its

flux density in these bands would be tabulated. However, concerns over possible

variability of this star, its possible IR excess, and the fact that it is not observable

from the Southern hemisphere, has led to modified approaches in which the star Sirius

is also taken as a calibrator and/or in which a model star is used instead. The latter

approach has been taken in Table 1.1 which lists the reference flux densities for

Table 1.1. Standard filters and magnitude calibrationa

U B V R I J H K L L

beff 0.366 0.438 0.545 0.641 0.798 1.22 1.63 2.19 3.45 3.80

c 0.065 0.098 0.085 0.156 0.154 0.206 0.298 0.396 0.495 0.588

f d01.790 4.063 3.636 3.064 2.416 1.589 1.021 0.640 0.285 0.238

f e0417.5 632 363.1 217.7 112.6 31.47 11.38 3.961 0.708 0.489

ZP 0.770 0.120 0.000 0.186 0.444 0.899 1.379 1.886 2.765 2.961

ZP 0.152 0.601 0.000 0.555 1.271 2.655 3.760 4.906 6.775 7.177

aUBVRIJHKL Cousins–Glass–Johnson system (Ref. [18]). The table values are for a fictitious A0 star which has0 magnitude in all bands. A star of flux density, f in units of 1020 erg s1 cm2 Hz1 or f in units of1011 erg s1 cm2 A1, will have a magnitude, m ¼ 2:5 logðfÞ 48:598 ZP or m ¼ 2:5 logðfÞ21:100 ZP, respectively. bThe effective wavelength, in m, is defined by eff ¼ ½

R f ðÞRW ðÞ d=

½Rf ðÞRW ðÞ d, where f ðÞ is the flux of the star at wavelength, , and RW ðÞ is the response function of

the filter in band W (see Figure 1.3). Thus, the effective wavelength varies with the spectrum of the starconsidered. cFull width at half-maximum (FWHM) of the filters in m. dUnits of 1020 erg s1 cm2 Hz1.eUnits of 1011 erg s1 cm2 A1.


reference magnitudes of zero in all filters. The flux density and reference flux density

must be in the same units. The above equations can be rewritten as,

m ¼ 2:5 logðfÞ 21:100 ZP

m ¼ 2:5 logðfÞ 48:598 ZP ð1:27Þ

where f is in units of erg cm2 s1 A1, f is in units of erg cm2 s1 Hz1, and ZP,

ZP are called zero point values (Ref. [18]). Example 1.4 provides a sample calculation.

Example 1.4

An apparent magnitude of B ¼ 1:95 is measured for the star, Betelgeuse. Determine its fluxdensity in units of erg cm2 s1 A1.

Table 1.2. Examples of apparent visual magnitudesa

Object or item Visual magnitude Comments

Sun 26.8

Approx. maximum of a supernova 15 assuming V ¼ 0 precursor

Full Moon 12.7

Venus 3.8 to 4.5b brightest planet

Jupiter 1.7 to 2.5b

Sirius 1.44 brightest nighttime star

Vega 0.03 star in constellation Lyra

Betelgeuse 0.45 star in Orion

Spica 0.98 star in Virgo

Deneb 1.23 star in Cygnus

Aldebaran 1.54 star in Taurus

Polaris 1.97 the North Starc

Limiting magnituded 3.0 major city

Ganymede 4.6 brightest moon of Jupiter

Uranus 5.7e

Limiting magnituded 6.5 dark clear sky

Ceres 6.8e brightest asteroid

Pluto 13.8e

Jupiter-like planet 26.5 at a distance of 10 pc

Limiting magnitude of HSTf 28.8 1 h on A0V star

Limiting magnitude of OWLg 38 future 100 m telescope

aFrom Ref. [71] (probable error at most 0.03 mag) and on-line sources. bTypical range over a year. cA variablestar. dThis is the faintest star that could be observed by eye without a telescope. It will vary with the individualand conditions. eAt or close to ‘opposition’ (180 from the Sun as seen from the Earth). f ‘Hubble SpaceTelescope’, from Space Telescope Science Institute on-line documentation. The limiting magnitude varieswith instrument used. The quoted value is a best case. gOWL (the ‘Overwhelmingly Large Telescope’) refersto the European Southern Observatory’s concept for a 100 m diameter telescope with possible completionin 2020.


From Eq. (1.26) and the values from Table 1.1, we have,

B B0 ¼ 1:95 0 ¼ 2:5 logfB

632 1011

ð1:28Þ

Solving, this gives fB ¼ 1:0 109 erg cm2 s1 A1 for Betelgeuse.Eq. (1.27) can also be used,

B ¼ 1:95 ¼ 2:5 logðfBÞ 21:100 þ 0:601 ð1:29Þ

which, on solving, gives the same result.

1.6.2 Absolute magnitude

Since flux densities fall off as 1r2, measurements of apparent magnitude between stars do

not provide a useful comparison of the intrinsic properties of stars without taking into

account their various distances. Thus the absolute magnitude has been introduced,

either as a bolometric quantity, Mbol, or in some waveband (e.g. MV, MB, etc). The

absolute magnitude of a star is the magnitude that would be measured if the star were

placed at a distance of 10 pc. Since the magnitude scale is relative, we can let the

reference star in Eq. (1.25) be the same star as is being measured but placed at a

distance of 10 pc,

mM ¼ 2:5 logf

f10 pc

¼ 5 þ 5 log

d

pc

ð1:30Þ

where we have dropped the subscripts for simplicity and used Eq. (1.9). Here d is the

distance to the star in pc. Eq. (1.30) provides the relationship between the apparent and

absolute magnitudes for any given star. The quantity, mM, is called the distance

modulus. Since this quantity is directly related to the distance, it is sometimes quoted as

a proxy for distance. Writing a similar equation for a reference star and combining with

Eq. (1.30) (e.g. Prob. 1.12) we find,

M Mb ¼ 2:5 logL

L

ð1:31Þ

where we have used the Sun for the reference star. Eq. (1.31) has been explicitly written

with bolometric quantities (Table G.3) but one could also isolate specific bands, as

before, provided the correct reference values are used.


1.6.3 The colour index, bolometric correction, and HR diagram

The colour index is the difference between two magnitudes in different bandpasses for

the same star, for example,

B V ¼ 2:5 logfB

fV

fV0

fB0

¼ 2:5 log

fB

fV

ðZPB ZPVÞ ð1:32Þ

or between any other two bands. Eq. (1.32) is derivable from Eqs. (1.26) or (1.27).

Various colour indices are provided for different kinds of stars9 in Table G.7. Since this

quantity is basically a measure of the ratio of flux densities at two different wavelengths

(with a correction for zero point), it is an indication of the colour of the star. A positive

value for B V, for example, means that the flux density in the V band is higher than that

in the B band and hence the star will appear more ‘yellow’ than ‘blue’ (see Table G.5).

The colour index, since it applies to a single star, is independent of distance. (To see

this, note that converting the flux density to a distance-corrected luminosity would

require the same factors in the numerator and denominator of Eq. (1.32)). Conse-

quently, the colour index can be compared directly, star to star, without concern for the

star’s distance. We will see in Sect. 4.1.3 that colour indices are a measure of the

surface temperature of a star. This means that stellar temperatures can be determined

without having to know their distances.

Since a colour index could be written between any two bands, one can also define an

index between one band and all bands. This is called the bolometric correction, usually

defined for the V band,

BC ¼ mbol V ¼ Mbol MV ð1:33Þ

For any given star, this quantity is a correction factor that allows one to convert from a

V band measurement to the bolometric magnitude (Prob. 1.16). Values of BC are

provided for various stellar types in Table G.7.

Figure 1.14 shows a plot of absolute magnitude as a function of colour index for over

5000 stars in the Galaxy near the Sun. Such a plot is called a Hertzsprung–Russell (HR)

diagram or a colour–magnitude diagram (CMD). The absolute magnitude can be

converted into luminosity (see Eq. 1.31) and the colour index can be converted to a

temperature (see the calibration of Figure G.1) which are more physically meaningful

parameters for stars. The HR Diagram is an essential tool for the study of stars and

stellar evolution and shows that stars do not have arbitrary temperatures and luminos-

ities but rather fall along well-defined regions in L T parameter space. The positions

of stars in this diagram provide important information about stellar parameters and

stellar evolution, as described in Sect. 3.3.2. It is important to note that the distance to

each star must be measured in order to obtain absolute magnitudes, a feat that has been

accomplished to unprecedented accuracy by Hipparcos, a European satellite launched

9Stellar spectral types will be discussed in Sect. 3.4.7.


in 1989. The future GAIA (Global Astrometric Interferometer for Astrophysics)

satellite, also a European project with an estimated launch date of 2011, promises to

make spectacular improvements. This satellite will provide a census of 109 stars and it

is estimated that it will discover 100 new asteroids and 30 new stars that have planetary

systems per day.

1.6.4 Magnitudes beyond stars

Magnitudes are widely used in optical astronomy and, though the system developed to

describe stars (for which specific intensities cannot be measured, Sect. 1.3), it can be

applied to any object, extended or point-like, as an alternate description of the flux

Figure 1.14. Hertzsprung–Russell (HR) diagram for approximately 5000 stars taken from theHipparcos catalogue. Hipparcos determined the distances to stars, allowing absolute magnitude tobe determined. The MV data have errors of about 0:1 magnitudes and the B V data have errors of< 0.025 magnitudes. Most stars fall on a region passing from upper left (hot, luminous stars) tolower right (dim, cool stars), called the main sequence. The main sequence is defined by stars thatare burning hydrogen into helium in their cores (see Sect. 3.3.2). The temperature, shown at thetop and applicable to the main sequence, was determined from the Temperature B V calibrationof Table G.7 and Figure G.1. The luminosity, shown at the right, was determined from Eqs. (1.31)and (1.33), using the bolometric correction from Table G.7. The straight line, defined by theequation, MV ¼ 8:58ðB VÞ þ 2:27, locates the central ridge of the instability strip for GalacticCepheid variable stars and the dashed lines show its boundaries (see Ref. [166a] and Sect. 5.4.2).


density. One could quote an apparent magnitude for other point-like sources such as

distant QSOs (‘quasi-stellar objects’)10, or of extended sources like galaxies. Since the

specific intensity of an extended object is a flux density per unit solid angle, it is also

common to express this quantity in terms of magnitudes per unit solid angle (Prob. 1.17).

1.7 Light aligned – polarization

The magnetic and electric field vectors of a wave are perpendicular to each other and to

the direction of propagation (Figure I.2). A signal consists of many such waves

travelling in the same direction in which case the electric field vectors are usually

randomly oriented around the plane perpendicular to the propagation direction. How-

ever, if all of the electric field vectors are aligned (say all along the z axis of Figure I.2)

then the signal is said to be polarized. Partial polarization occurs if some of the waves

are aligned but others randomly oriented. The degree of polarization, Dp is defined as

the fraction of total intensity that is polarized (often expressed as a percentage),

Dp Ipol

Itot

¼ Ipol

Ipol þ Iunpol

ð1:34Þ

Polarization can be generated internally by processes intrinsic to the energy generation

mechanism (see Sect. 8.5, for example), or polarization can result from the scattering

of light from particles, be they electrons, atoms, or dust particles (see Sect. 5.1 or

Appendix D). When polarization is observed, Dp is usually of order only a few percent

and ‘strong’ polarization, such as seen in radio jets (Figure 1.4), typically is

Dp < 15 per cent (Ref. [175]). Values of Dp over 90 per cent, however, have been

detected in the jets of some pulsars (Ref. [113]) and in the low frequency radio

emission from planets (Sect. 8.5.1). For the Milky Way, Dp ¼ 2 per cent over distances

2 to 6 kpc from us due to scattering by dust (Ref. [175]). In practical terms, this means

that polarized emission is usually much fainter than unpolarized emission and requires

greater effort to detect.

Problems

1.1 Assuming Cas A (see Sect. 1.1 for more data) has a spectral luminosity between

1 ¼ 2 107 Hz and 2 ¼ 2 1010 Hz, of the form, L ¼ K 0:7, where K is a constant.

Determine the value of K, and of L and L at ¼ 109 Hz. What are the units of K?

1.2 Find, by comparison with exact trigonometry, the angle, (provide a numerical value

in degrees), above which the small angle approximation, Eq. (B.1), departs from the exact

10A QSO is the bright active nucleus of a very distant galaxy that looks star-like at optical wavelengths. QSOs thatalso emit strongly at radio wavelengths are called quasars.

1.7 LIGHT ALIGNED – POLARIZATION 25

result by more than 1 per cent. How does this compare with the relatively large angle

subtended by the Sun?

1.3 Determine the flux density (in Jy) of a cell phone that emits 2 mW cm2 at a

frequency of 1900 MHz over a bandwidth of 30 kHz, and of the Sun, as measured at the

Earth, at the same frequency. (Eq. 4.6 provides an expression for calculating the specific

intensity of the Sun.) Compare these to the flux density of the supernova remnant, Cas A

( 1900 Jy as measured at the Earth at 1900 MHz) which is the strongest radio source in the

sky after the Sun. Comment on the potential of cell phones to interfere with the detection of

astronomical signals.

1.4 (a) Consider a pulsar with radiation that is beamed uniformly into a circular cone of

solid angle, . Rewrite the right hand side (RHS) of Eq. (1.9) for this case.

(b) If ¼ 0:02 sr, determine the error that would result in L if the RHS of Eq. (1.9) were

used rather than the correct result from part (a).

1.5 Determine the percentage variation in the solar flux incident on the Earth due to its

elliptical orbit. Compare this to the variation shown in Figure 1.6.

1.6 Determine the flux in a perfectly isotropic radiation field (i.e. I constant in all

directions).

1.7 (a) Determine the flux and intensity of the Sun (i) at its surface, (ii) at the mean

distance of Mars, and (iii) at the mean distance of Pluto.

(b) How large (in arcmin) would the Sun appear in the sky at the distances of the two

planets? Would it appear resolved or as a point source to the naked eye at these locations?

That is, would the angular diameter of the Sun be larger than the resolution of the human

eye (Table G.5) or smaller? (Sects. 2.2, 2.2.3, and 2.3.2 provide more information on the

meaning of ‘resolution’.)

1.8 The radio spectrum of Cas A, whose image is shown in Figure 1.2, is given in

Figure 8.14 in a log–log plot. The plotted specific intensity can be represented by,

I ¼ I0 ð=0Þ in the part of the graph that is declining with frequency, where 0 is any

reference frequency in this part of the plot, I0 is the specific intensity measured at 0 and is the slope. In Prob. 1.1, we assumed that ¼ 0:7. Now, instead, measure this value from

the graph and determine, for the radio band from 1¼2 107 to 2 ¼ 2 1010 Hz,

(a) the intensity of Cas A, I,

(b) the solid angle that it subtends in the sky, ,

(c) its flux, f ,

(d) its radio luminosity, Lrad. Confirm that this value is approximately equal to the value

given in Sect. 1.1.


1.9 On average, the brightness of the Whirlpool Galaxy, M 51 (see Figure 3.8 or 9.8),

which subtends an ellipse of major minor axis, 11.20 6.90 in the sky, is 2.1 times that of

the Andromeda Galaxy, M 31 (subtending 1900 600). What is the ratio of their flux

densities?

1.10 Where does the Solar System end? To answer this, find the distance (in AU) at which

the radiation energy density from the Sun is equivalent to the ambient mean energy

density of interstellar space, the latter about 1012 erg cm3. After 30 years or more of

space travel, how far away are the Pioneer 10 and Voyager 1 spacecraft? See http://

spaceprojects.arc.nasa.gov/Space_Projects/pioneer/PNhome.html

and http://voyager.jpl.nasa.gov. Are they out of the Solar System?

1.11 Consider a circular, perfectly reflecting Solar sail that is initially at rest at a distance

of 1 AU from the Sun and pointing directly at it. The sail is carrying a payload of 1000 kg

(which dominates its mass) and its radius is Rs ¼ 500 m.

(a) Derive an expression for the acceleration as a function of distance, aðrÞ, for this Solar

sail. Include the Sun’s gravity as well as its radiation pressure. (The constants may be

evaluated to simplify the expression.)

(b) Manipulate and integrate this equation to find an expression for the velocity of the

Solar sail as a function of distance, vðrÞ. Evaluate the expression to find the velocity of the

Solar sail by the time it reaches the orbit of Mars.

(c) Finally, derive an expression for the time it would take for the sail to reach the orbit

of Mars. Evaluate it to find the time. Express the time as seconds, months, or years,

whatever is most appropriate.

1.12 Derive Eq. (1.31) (see text).

1.13 Repeat Example 1.4 but expressing the flux density in its frequency-dependent form.

1.14 For the U band and L band filters, verify that f corresponds to f in Table 1.1. Why

might there be minor differences?

1.15 Refer to Table 1.2 for the following.

(a) Determine the range (ratio of maximum to minimum flux density) over which the

unaided human eye can detect light from astronomical objects. Research the range of

human hearing from ‘barely audible’ to the ‘pain threshold’ and compare the resulting

range to the eye.

(b) Use the ‘Multiparameter Search Tool’ of the Research Tools at the web site of the

Hipparcos satellite (http://www.rssd.esa.int/Hipparcos/) to determine what

percentage of stars in the night sky one would lose by moving from a very dark country site

into a nearby light polluted city.

1.7 LIGHT ALIGNED – POLARIZATION 27

(c) The star, Betelgeuse, is at a distance of 130 pc. Determine how far away it would

have to be before it would be invisible to the unaided eye, if it were to undergo a supernova

explosion.

1.16 A star at a distance of 25 pc is measured to have an apparent magnitude of V ¼ 7:5.

This particular type of star is known to have a bolometric correction of BC ¼ 0:18.

Determine the following quantities: (a) the flux density, fV in units of erg cm2 s1 A1,

(b) the absolute V magnitude,MV, (c) the distance modulus, (d) the bolometric apparent and

absolute magnitudes, mbol, and Mbol, respectively, and (e) the luminosity, L in units of L.

1.17 A galaxy of uniform brightness at a distance of 16 Mpc appears elliptical on the sky

with major and minor axis dimensions, 7:90 1:40. It is observed first in the radio band

centred at 1.4 GHz (bandwidth ¼ 600 MHz) to have a specific intensity of 4.8 mJy beam1,

where the ‘beam’ is a circular solid angle of diameter, 1500 (see also Example 2.3d). A

measurement is then made in the optical B band of 22.8 magnitudes per pixel, where the

pixel corresponds to a square on the sky which is one arcsecond on a side. Determine (all in

cgs units) I , f , f , and L of the galaxy in each band. In which band is the source brighter? In

which band is it more luminous?

1.18 The limiting magnitude of some instruments can be pushed fainter by taking

extremely long exposures. Estimate the limiting magnitude of the Hubble Ultra-Deep

Field from the information given in Figure 3.1.


2Measuring the Signal

All these facts were discovered . . . with the aid of a spyglass which I devised, after first being

illuminated by divine grace.

–Galileo Galilei in The Starry Messenger

2.1 Spectral filters and the panchromatic universe

The first astronomical instrument was the human eye. From prehistoric times until

today, human beings have surveyed the heavens with this most elegant and effective

‘telescope’. The eye, like any other instrument, acts like a filter, accepting light at some

wavelengths and filtering out others. The spectral response function of the human eye

is shown in Figure 2.1 (see also Table G.5) and illustrates the relative ability of the eye

to detect light at different wavelengths in the visual part of the electromagnetic

spectrum1. If an incoming signal were uniformly bright across all wavelengths, the

eye would not perceive it that way, but rather would see light in the range 500–550 nm

(depending on conditions) as the brightest. Outside of this relatively narrow visual

band, the eye is unable to detect a signal at all.

Attempts have been made to improve upon the human eye for at least 700 years.

Early versions of eyeglasses, for example, date to 1284 or 1285 AD and possibly

earlier in a more primitive fashion. However, it was not until Galileo first turned a

telescope on the Moon, the Sun and planets in the early 17th Century, that combina-

tions of lenses were put towards the purpose of astronomical observation. Since

Galileo, we have enjoyed 400 years of increasingly larger and more sophisticated

optical telescopes. By contrast, other wavebands were not even known to exist until


1‘Visual’ will be used to indicate the part of the spectrum to which the human eye is sensitive, whereas ‘optical’will refer to the part of the spectrum to which ground-based optical telescopes are sensitive; the latter includes thenear-IR and the near-UV.

the discovery of the infrared by William Herschel in 1800. Once it was understood

that there could be emission invisible to the human eye (see Table G.6), it was only a

matter of time before a variety of developing technologies provided the means to

detect and form images in non-visual wavebands. The technology to detect radio

signals, for example, matured rapidly after World War II. In a sense, such technol-

ogies provide us with eyeglasses to enhance the response of the human eye ‘sideways’

into other wavebands.

One other important constraint inhibited the discovery of far-IR, UV, X-ray and

-ray emission from astronomical objects until the latter-half of the 20th Century – that

of the Earth’s atmosphere. The spectral response function of the atmosphere is shown

in Figure 2.2 and reveals that only in the optical and radiowavebands is the atmosphere

sufficiently transparent for ground-based observations. The 1957 launch of Sputnik by

the former Soviet Union marked the dawn of the space-age and, with it, a rich era of

astronomical discovery. This era continues today, with space-based probes realising

increasingly sensitive and highly detailed images of astronomical objects at the

previously inaccessible ‘invisible’ wavelengths.

Thus, the history of astronomy has largely been driven by two spectral response

functions: that of the human eye and that of the Earth’s atmosphere. Our knowledge of

the Universe is still based on information obtained through a strong optical and ground-

based bias. In 2005, for example, there were 45 optical research telescopes greater than

2.4 m in diameter and approximately 17 more in either the construction or planning

stages. Only one of these (the Hubble Space Telescope) was not ground based. In the

same year, about 17 space-based astronomical telescopes were in operation

with another 17 planned or proposed. The amount and quality of non-optical and

Figure 2.1. Response function of the human eye for photopic (daylight) and scotopic (dark-adapted) conditions. The peak wavelengths are indicated. The dark-adapted response is dominatedby rods in the eye’s retina whereas daylight vision is dominated by the cones. Sensitivity to light ismuch higher under dark-adapted conditions, as shown, but the ability to distinguish colour ismarkedly diminished. Related information can be found in Table G.5

30 CH2 MEASURING THE SIGNAL

space-based data are rapidly increasing, however2, and are revealing aspects of our

Universe never before seen, or even contemplated.

Why is such a concerted effort required over so many wavebands? Figure 2.3, which

shows the spectrum of the galaxy, NGC 2903, provides a good illustration. This rather

normal spiral galaxy (see Figure 2.11 for an optical image) emits over a wavelength

range spanning ten orders of magnitude, and probably more if the observations were

available. Awealth of information is hidden in this plot with each band revealing new

and different insights about the source. If an image were taken at each waveband, the

images would not be identical to each other. Further clues about the system are

uncovered by this changing appearance with waveband. It is clearly impossible to

truly understand the galaxy with observations in the optical band alone – an approach is

required that is ‘panchromatic’, or spanning all wavebands. If we now consider the

wide range of objects one could observe in the Universe besides normal galaxies, the

imperative to widen our waveband horizons becomes even greater.

Astronomy is now in a discovery era, driven by technological improvements in the

telescope and its concomitant instruments. It is therefore essential to have some

understanding of how this technology, whether space-based or under the blanket of

–9

–8

–7

–6

–5

–4

–3

–2

–1

06 8 10 12 14 16 18 20 22 24

0

10

20

30

40

5060

70

80

90

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110120130140150

1086420–2–4–6

log (photon energy/eV)

log (frequency/Hz)

log

(fra

ctio

n of

atm

osph

ere)

Alti

tude

, km

Figure 2.2. Atmospheric transparency curve for the Earth. The altitude (right hand side) indicateshow high one would have to put a telescope in order for the atmosphere to be transparent at thatfrequency. This is expressed as a fraction of the total atmosphere on the left hand side. In this plot,wavelength decreases to the right in contrast to Figure 2.1. The two atmospheric windows are in theradio and optical parts of the spectrum. Most atmospheric absorption is due to water vapour (H2O),then carbon dioxide (CO2), and then ozone (O3). The single spike around 60 GHz is due to oxygen(O2). Adapted from Ref. [101]

2For a good sampling of the number of ground-based and space-based facilities available, see the website of theRoyal Astronomical Society http://www.ras.org.uk under astrolinks.

2.1 SPECTRAL FILTERS AND THE PANCHROMATIC UNIVERSE 31

our atmosphere, collects, filters, distorts, and ultimately reveals the secrets of the

astronomical sources that so intrigue us.

2.2 Catching the signal – the telescope

Although the technology required to collect light and the quality of the result vary

enormously across wavebands, the basic principles are the same: the job of a telescope

is to collect as much light as possible and to focus it on a detector3, forming an image, if

possible. For scientific purposes, there are two main ingredients in this process: a

surface that collects and focusses the light called the objective (also called the primary

lens, or mirror or antenna, depending on what is used), and a device (the detector) that

detects and converts the received signal into some form, usually digital, for storage and

analysis via computer. This is analogous to what the human eye does. The pupil is the

opening through which light can pass, acting as the collector, focussing is achieved by

the lens and the interior fluid, and the detector is the retina with its plethora of rods and

cones. The eye’s detector then converts the light into electrical signals which are sent to

the brain to be analysed. A telescope, therefore, acts rather like a giant eye.

Figure 2.3. Spectrum of the galaxy, NGC 2903 (Ref. [105]). The frequency is shown at the bottomand some specific wavebands are labelled at the top. See Sect. 10.1.1 for a discussion of thisspectrum. (Adapted from the NASA/IPAC Extra galactic Database)

3Focussing is not possible, however, at the highest -ray energies (see Figure 2.7.b).


Figure 2.4 provides a diagram of a simple telescope that uses a lens as the objective.

However, the results, below, are the same for a reflecting instrument. Since astronom-

ical signals are at a great distance, the incoming wavefronts from a source at any angle

in the sky will be plane parallel. This means that light rays that trace the direction of the

wave front are also parallel to each other, and are intercepted by the entire aperture, as

shown. It also means that the position of the focussed source behind the lens is the

actual focal length of the lens, f , and this is where the detector must therefore be placed.

The geometry of the diagram shows how the angular size of any object on the sky, , isrelated to a linear size on the detector, l,

tan ¼ l

fð2:1Þ

where (radians) is presumed small. Thus telescopes with longer focal lengths result in

larger images on the detector. Since f is a constant for any telescope, so is the quantity,

=l, sometimes called the plate scale for optical instruments.

Telescopes are designated by their diameters and focal ratios, the latter being the

ratio of the focal length to the diameter of the objective. For example, a 3 m f/5

telescope would be 3 m in diameter and have a focal length of f ¼ 5 3 ¼ 15m.

Bigger telescopes can detect fainter signals (Sect. 2.2.1) and also result in images with

better clarity or finer detail, also called higher resolution (Sect. 2.2.4). However, the

latter is subject to other constraints which will be outlined in Sects. 2.2.3, and 2.3.2.

Objective lens

Rays frompoint A

A

B θpix

θ pixθpix

A

B CCD

Figure 2.4. Diagram of a simple optical telescope, showing the relationships between the focallength, the linear scale on the detector, and the angular scale in the sky. Note that the angles havebeen exaggerated. The rays from a very distant point, A, all impinge upon the objective parallel toeach other and at the same angle. The same is true of the rays from point B. The rays shown arethose passing through the centre of the lens which experience no net bending due to lenssymmetry. If visual observing is desired, it would be necessary to place a lens (called the ocular oreyepiece) in the focal plane of the telescope instead of the detector. The ocular would then bendthe light again so that a human being can view the image.

2.2 CATCHING THE SIGNAL – THE TELESCOPE 33

2.2.1 Collecting and focussing the signal

At optical wavelengths, collecting and focussing light is done via either a lens

(refracting the light) or curved mirror (reflecting the light). For everything else

equal, larger collecting areas result in brighter images on the detector because a

larger area will intercept more of the source luminosity. Since astronomical sources

are typically very faint, telescopes are built as large as costs and mechanical

structures allow. Large optical telescopes used for research always use mirrors,

since they require finishing only on one side and can be supported more easily

with the mirror at the bottom of the telescope tube. Moreover, since reflection affects

all wavelengths of optical light the same way whereas refraction bends light

differently for different wavelenghts (e.g. see the Appendix to this chapter) the

use of mirrors avoids certain aberrations associated with lenses4. Since the light then

reflects upwards, either a detector must be placed at the focal point high up in the

tube, called the prime focus, or else another secondary mirrormust be used to reflect

the signal back down again to a detector near the bottom (Figure 2.5). If the incoming

4In particular, lenses are subject to chromatic aberration because of wavelength-dependent refraction, which putsthe focal point of blue light closer to the lens than the focal point of red light.

Classical Cassegrain

F

Source

Figure 2.5. A distant point source emits its light in all directions, but the wavefront is planeparallel by the time it reaches the telescope. Rays denote the direction of the incoming planeparallel wavefront, in this case from a point source ‘on-axis’, i.e. at the centre of the field of view.This telescope shows a typical design in which curved mirrors collect and focus the light onto adetector behind the primary mirror. Such a design may have different names depending on thecurvature of the mirrors and whether additional corrective lenses are used


light changes direction via mirrors, the light path is said to be folded. Modern

detection instruments can be rather massive and so are usually placed at or near

the bottom of the telescope. A variety of light paths are possible, however, including

those with secondary mirrors at positions that do not block the aperture and those

with detectors on platforms adjacent to the telescope tube.

Single-dish radio telescopes, infrared, optical, and ultraviolet telescopes all operate

similarly, with collection and focussing achieved via a primary reflector. Figure 2.6

shows a radio and optical telescope example.

To maintain the integrity of the signal, irregularities on the surface of the primary

reflector should be a small fraction (e.g. 1/10th if possible) of a wavelength. Thus

optical telescopes require finely engineered smooth mirrors whereas a wire mesh may

suffice at radiowavelengths. At X-ray wavelengths, however, evenvery smoothmirrors

have rather limited focussing capabilities since X-rays tend to be absorbed, rather than

reflected, when hitting a mirror perpendicularly. Instead X-rays are focussed via a

series of glancing reflections (Figure 2.7.a)5. At even higher energies, -rays cannot befocussed at all and other techniques, such as tracing the signal direction through

multiple detectors, must be used to determine the direction in the sky from which

the light has come. (Figure 2.7.b).

5For X-ray telescopes, the ability of the telescope to focus onto a detector is what limits the spatial resolution.

Figure 2.6. (a) The Arecibo Telescope, which operates at radio wavelengths, has a primaryreflector 305 m in diameter. Suspended approximately 137 m above the centre of the dish at itsprime focus is a platform containing several more small reflectors and the detectors. (Photocourtesy of the NAIC – Arecibo Observatory, a facility of the NSF.) (b) A view of Gemini North, an8.1 m diameter optical telescope situated on Mauna Kea in Hawaii. Light coming down the opentube reflects back from the primary mirror to a secondary mirror which reflects the light againthrough a hole in the primary to the detectors below. The effective focal length is 128.12 m.(Reproduced by permission of the Gemini observatory/AURA)


2.2.2 Detecting the signal

The detector is a device for turning the collected light into another form that can be

analysed. A wide variety of detectors are used depending on waveband and desired

output: photographic films, radio receivers, microchannel plates6, CCDs (see below)

and many others.

The simplest detector is one that accepts a signal from a single position on the sky.

The telescope is pointed at a position on the sky and the detector records values

(usually a spectrum, that is, the emission as a function of wavelength or frequency) that

correspond to only this one position. If a map of an extended source is desired, the

telescope must re-point to another position on the source, the detector makes a

recording again, and so on until a picture of the source is built up consecutively.

Many radio and mm-wave telescopes, which have complex receiver systems, still work

on this principle. However, this process is slow and, wherever possible, has been

replaced by imaging detectors which consist of many ‘picture elements’ or pixels. In

general, imaging detectors are called focal plane arrays.

For optical work, an example of a focal plane array is the photographic plate which

has been widely used in the past and is still sometimes used today, especially when very

large fields of view are desired or if images, rather than numerical scientific results, are

desired. However, the photograph has been virtually universally replaced for scientific

work by theCharge Coupled Device, orCCD (Figure 2.8.a), which is a semi-conductor

detector (a ‘chip’) like those used in digital cameras. The CCD’s high sensitivity to low

light levels, its linear response to increasing light levels7, and its direct interface to

computers make it the ideal detector for most scientific purposes. Larger format CCDs

are also now becoming available, making earlier limitations of small fields of view less

problematic. In each pixel of the CCD, electrons are released when exposed to light in

Figure 2.7. (a) A schematic view of the space-based Chandra X-ray telescope, showing howsuccessive grazing reflections are used to focus the light onto a detector. (Reproduced bypermission of NASA/CXC/SAO) (b) Artist’s drawing of the Gamma-ray Large Area Space Telescope(GLAST). This telescope is basically a large cube in space with no mirrors or tubes. The detectoritself has built-in capabilities to measure the directions and energies of incident gamma rays.GLAST is a joint European–American–Japanese project with a proposed launch date in 2007.(Reproduced by permission of NASA E/PO, Sonoma State University)

6These are detection and amplifying devices used to convert a single high energy photon, such as an X-ray, intomany electrons at the output.7Any given pixel, however, will become saturated if exposed to too high a level.


some energy range for which the CCD has been designed. This is essentially the

photoelectric effect. The number of electrons released is proportional to the number of

photons incident. The electrons remain at the location of the pixel during the exposure

and are read out at the end, line by line, via an applied voltage. The information is then

stored on the computer in a file that represents it as a two-dimensional numerical array

with array locations corresponding to the location on the chip, higher numbers

representing brighter light.

Multi-pixel detectors in the focal plane of the telescope are used across all wave-

bands from radio to X-rays, with the CCD replaced by whatever detector is required for

the given band. At radio wavelengths, for example, the pixels consist of feedhorns

which guide the signal into other electronic equipment. Figure 2.8.b, illustrates this,

showing the 7 pixel focal plane array used at the Arecibo Radio Telescope.

2.2.3 Field of view and pixel resolution

Since any detector has a finite pixel size, lpix, and a finite diameter, ldet, Eq. (2.1)

indicates that these limits will impose corresponding limits on the angular resolution,

pix, also called the pixel field of view, and the angular field of view, FOV, respectively.Detectors with many small pixels have the potential to produce higher resolution

images, provided there are no other limitations on the resolution (see next sections).

For example, a photographic emulsion with tiny grains will show more detail than one

with large grains. An illustration of an image on a CCD in which the resolution is

Figure 2.8 (a) A CCD camera like this one can be affixed to the back of an optical telescope toobtain images. Most of the weight of the camera is in supportive systems such as the coolingsystem. The CCD itself (upper left inset) is quite small. This particular CCD is 2.46 cm on a side with1024 1024 pixels in the array, each of which is square and 24m on a side. (Reproduced bypermission of Finger Lakes Instrumentation) (b) By contrast, the Arecibo L-Band Feed Array (ALFA)weighs 600 kg and fills a small laboratory. When placed at the focal plane of the telescope, its sevencircular pixels of diameter 25 cm sample seven nearby regions on the sky with small gaps inbetween. (Reproduced by permission of CSIRO Australia. Credit: David Smyth)


limited by a large pixel size is shown in Figure 2.11.c. A previous example, showing a

single star smaller than the pixel field of view was given in Figure 1.9. As for the

angular field of view, physically larger detectors are able to accept light from a larger

range of angles, provided the telescope tube itself does not interfere.

2.2.4 Diffraction and diffraction-limited resolution

Since the telescope objective, however large it might be, collects only a portion of the

light from any source that illuminates the Earth, it is like a circular aperture which

accepts light that falls within it and rejects light that does not. This is analogous to a

laboratory situation in which light passes through a small hole and is subsequently

diffracted. Diffraction is the net result of interferencewithin the aperture. Each point on

any plane wavefront can be thought of as a new source of circular waves. This is known

asHuygen’s Principle. If there were no barrier (an infinite aperture), then the net result

of these circular waves interfering with each other would still be a plane wave.

However, with the barrier in place, only the points within the aperture interfere with

each other and the net result is a wave that fans out in a circular fashion at the edges.

The resulting diffraction pattern falling on a screen (or on a detector in the case of a

telescope) is shown in Figure 2.9 for a narrow slit. For small angles and a circular

aperture, the angular distance of the first null from the central point, N, and the full

width of the central peak at half maximum intensity (FWHM), FWHM, are, respectively,

N ¼ 1:22=D FWHM ¼ 1:02=D ð2:2Þ

where is the wavelength of the light andD is the diameter of the aperture.D could be

somewhat less than the physical size of the aperture if the light path is partially blocked

Figure 2.9. A laboratory aperture þ screen is analogous to a telescope þ detector. (a) Light isaccepted only through a narrow slit and projected onto a screen some distance away. A diffractionpattern results from interference within the opening. (b) A plot of intensity as a function of angle,, across the screen yields this pattern.


or otherwise imperfect. In such a case, D must be replaced by the effective aperture,

Deff . It is clear from Eq. (2.2) that if wewant a crisp small image of a point source, then

a larger telescope (larger D) is required. It is also worth remembering that higher or

larger corresponds to lower resolution.

A face-on view of the diffraction pattern in two dimensions for a circular aperture is

shown in Figure 2.10.a. This is the spatial response function of the telescope to a point

source that is evenly illuminating the aperture. Most of the emission falls within the

central peak called the Airy disk in optical astronomy and called the main lobe or main

beam in radio astronomy. However, a small amount of emission falls within the other

surrounding peaks or diffraction rings (sidelobes in radio astronomy). For uniform

aperture illumination, the Airy disk contains 84 per cent of the power falling on the

detector. The peak of the first diffraction ring occurs at a level 1.7 per cent of the central

peak and contains 8.6 per cent of the power, while the values for the tenth ring are 103

and 0.2, respectively (Ref. [178]). Thus, a point source in the sky will not form a point

source on the detector, but rather a diffraction pattern in which the light is spread out as

illustrated.

It is sometimes possible to modify the response function of the telescope. For

example, the power going into the diffraction rings could be reduced by weighting

the aperture so that it accepts more light towards the centre in comparison to the edges.

However, such weighting has the effect of reducing the effective size,D, of the aperture

thereby worsening its resolution. The opposite is also true. ‘Super-resolution’ can be

achieved by weighting the outer parts of the dish or mirror but only by increasing the

power going into the diffraction rings8.

Figure 2.10. (a) The two-dimensional diffraction pattern from a circular aperture illuminated bya single point source at infinity. The bright central spot is called the Airy disk. (b) Two closelyspaced point sources in the sky would create a pattern like this on the detector.

8In the limit, one could split up the aperture into many small elements and spread these elements out over a muchlarger area, effectively increasing D to a very large value. The signal can be reconstructed by analysing the cross-correlation between the various elements. This is the principle of interferometry commonly used in radioastronomy. The technique is increasingly used in optical astronomy as well, though it is more technicallydemanding at the smaller wavelengths. Another way to weight an aperture is to apply a mask which blocks lightfrom certain regions.


If a second star is now present, displaced from the first by some angle, , then two

sets of parallel light rays evenly illuminate the aperture, separated by this angle (see,

e.g. points A and B in Figure 2.4). Two diffraction patterns will now be on the

detector with their centres separated by . If these stars were separated from each

other by smaller and smaller angles in the sky, the diffraction patterns on the detector

would become progressively closer together, overlapping (as shown in

Figure 2.10.b) and finally merging. There is a minimum angle on the sky, d, atwhich it is still possible to just distinguish that there are two sources rather than one.

This limit is called the diffraction-limited resolution of the telescope, or just the

diffraction limit, and provides a more technical definition of the concept of resolu-

tion. For a circular aperture, the resolution is often specified by the Rayleigh

criterion which states that this will occur when the maximum of one image is placed

at the first minimum (first null) of the other. In practise, separation by the FWHM is

an adequate criterion and, as a rule of thumb, the diffraction-limited resolution of the

telescope may be found from,

d =D ð2:3Þ

The brightness distribution of an astronomical source can be thought of as a set of

many individual closely spaced point sources of different brightness. On the

detector, each point in the image will have superimposed on it the diffraction pattern

of a point source. Mathematically, this can be represented as a convolution (see

Appendix A.4) of the brightness distribution of the source with the diffraction

pattern of the telescope. The broader the central lobe of the diffraction pattern,

the poorer, or lower is the resolution and the source will appear as if out of focus.

Such a comparison is shown between Figure 2.11.a and Figure 2.11.b. Since the

Figure 2.11. Illustration of resolution for the spiral galaxy, NGC 2903 (distance, D ¼ 8:6 Mpc).The disk of this galaxy is circular, appearing as an ellipse in projection, with the minor axis of theellipse subtending the angle, 6.00, and the major axis, 12.60, on the sky. (a) A high resolutionimage such as might result from using a large aperture telescope, (b) a low resolution image suchas from a telescope with a small aperture, or from a large aperture telescope with poor seeing and(c) a low resolution image due to large pixels on the detector.


light in the outer diffraction rings is much lower in intensity than the main lobe,

these rings do not show up in Figure 2.11.b.

Lower resolution, whether due to telescope diffraction or other limitations

(Sect. 2.3.2), has the effect of spreading out the signal and lowering its peak

(note that the flux is preserved, cf. Figure 1.9). In the limit, if the resolution is

very low, the angular size of the source, S, becomes much less than d and the

source is said to be unresolved as if it were a point source (Prob. 2.8). Its apparent

angular size will then equal the angular size of the central lobe of the beam or Airy

disk.

2.3 The corrupted signal – the atmosphere

Any observation from the Earth’s surface must contend with the atmosphere. Not only

does the signal pass through this scattering, absorbing and emitting layer, but the effect

that the atmosphere has on a signal is strongly wavelength-dependent and is also

variable with time and with local conditions. Details of scattering and absorption will

be discussed more fully in Chapter 5 but herewe consider the atmosphere in terms of its

undesirable effects on observations.

The spectral response curve of the atmosphere (Figure 2.2) indicates that there

are only two atmospheric windows for which ground-based observations are pos-

sible, the radio and the optical. Of these, the most transparent band is the radio.

Radio wavelengths are large in comparison to the typical size of atmospheric

particles (atoms, molecules, and dust particles) so the probability of interaction

with these particles is small and atmospheric effects are mostly negligible. This

means that observations in the radio band can be made day or night, during cloud

cover or clear weather and from sea level, if desired. The sharp cutoff at very low

radio frequency ( 10 MHz, Figure 2.2) is due to the ionosphere which reflects,

rather than transmits incoming waves (see Appendix E). The ionosphere is an

ionised layer ranging from a height of 50 km to 500 km with a maximum electron

density (the number density of free electrons) of between 104 and 106 cm3,

depending on conditions. At the high frequency end of the radio band

(Z 300GHz; [ 0:1 cm, Figure 2.2) the atmosphere also becomes more impor-

tant and radio telescopes operating in this range must be placed on high mountains

and are put in domes just like optical telescopes, as Figure 2.12 illustrates.

Techniques such as beam switching or chopping are also employed at these as

well as IR wavelengths, which involve rapidly switching the light path back and

forth from the source to a nearby region of blank sky. The image of the source is

then corrected for time variable atmospheric emission by subtracting off the nearby

sky brightness distribution which closely corresponds in time.

The optical window (which includes the near IR and near-UV) is strongly affected by

the atmosphere and therefore optical research telescopes are placed at high altitudes

above as much of the ‘weather’ as possible. Some specific effects of the atmosphere on

an optical signal are discussed below.

2.3 THE CORRUPTED SIGNAL – THE ATMOSPHERE 41

2.3.1 Atmospheric refraction

The bending of light as it passes from one type of medium to another is described by

Snell’s Law (Table I.1). Such bending occurs as light passes from space into denser and

denser sections of the atmosphere of the Earth. The bending increases systematically

with zenith angle which is the angle between the zenith and the source, making the

source appear higher in the sky than it actually is. Extended objects that span a range of

zenith angle will therefore also appear distorted, as does the Sun when it is close to the

horizon at sunrise or sunset (Figure 2.13, Prob. 2.2). Thus, models accounting for

known systematic refraction must be built into telescope control software in order to

accurately point the telescope to a source in the sky. The details of atmospheric

Figure 2.12. The James Clerk Maxwell Telescope, operating at sub-mm wavelengths, is located onMauna Kea in Hawaii at an elevation of 4200 m. Although this is a radio telescope, it is housed in adome and, under normal operating conditions, the dome opening is covered by a protective clothmembrane that is transparent at the operating wavelengths. Contrast this image to the Areciboradio telescope that operates at longer wavelengths, shown in Figure 2.6.a, which is at an altitudeof only 497 m. (Reproduced by permission of Robin Phillips)

Figure 2.13. The Sun appears slightly flattened at sunrise or sunset.


refraction at optical wavelengths are provided in the Appendix at the end of this

chapter9. A more general discussion of refraction is given in Sect. 5.3.

2.3.2 Seeing

The atmosphere is turbulent and this turbulence can be represented by ‘cells’ of gas that

are constantly in motion. Many such cells with different sizes and slightly different

temperatures may be present above the telescope aperture. A good approximation to

atmospheric turbulence is one in which the pressure remains constant but the tem-

perature fluctuates, resulting in fluctuations in the index of refraction (see Eq. 2.A.4).

This introduces a small random element of refraction superimposed on the systematic

refraction discussed above. The net result is that an incoming wave that is originally

plane parallel will become distorted, and the tilt of thewavefront results in a focal point

on the detector that is shifted slightly from the non-tilted position. If there are multiple

cells above the aperture, then there will be multiple distorted images (each convolved

with the telescope’s diffraction pattern) on the detector (Figure 2.14). Each individual

image is called a speckle. The larger the aperture, the more cells can be in front of it

and the more speckles there will be on the detector. The largest effects are due to cells

near the ground that are blown across the aperture by wind. The sizes of these

atmospheric cells vary, but a typical value is r0 10 cm in the visual, though it can

be up to 20 cm at a high altitude where there is more atmospheric stability.

Since atmospheric cells are constantly in motion, the speckle pattern changes rapidly

with time. The timescale over which significant changes in the pattern occur is of order

the time it takes for a cell to move across the aperture,

t ¼ ar0

vð2:4Þ

where v is the wind speed and the constant, a, has a value of 0:3. For a wind speed ofv ¼ 300 cm s1 (11 km h1) and r0 ¼ 10 cm, therefore, this timescale is 10 ms. Thus,

to see diffraction limited images using a large aperture telescope, it is necessary to take

very rapid exposures shorter than 10 ms in duration. This poses a problem, however,

because exposures shorter than 10 ms are generally not long enough to produce a

significant response on the detector except for the very brightest stars.

In order for the detector to register a signal from a faint source, it is necessary to take

exposures that are much longer than 10 ms. During the exposure, the shifting speckles

build up a signal on the detector, filling in the gaps and forming a smeared out image over

a small region (Figure 2.14). The characteristic angular size of this region, s, taken to bethe FWHM, is called the seeing disk or just the seeing. Seeing, therefore, refers to the

change in angular position of the signal, or the change in the phase of thewavefront, with

time due to the atmosphere. The apparent position of the star in the sky is at the centre of

9Refraction exists at radio wavelengths as well and is similar in magnitude to the value for the near-IR.


its seeing disk. Seeing varies with wavelength, and with altitude and geographic

location. It also varies from night to night and can change over the course of the night.

Thus, the resolution of a large optical ground-based telescope is limited, not by

diffraction, d, but by the seeing, s, since over the timescale of a typical exposure, the

stellar image will smear out over a size, s. Large research telescopes have historicallybeen designed to collect more light, allowing fainter sources to be detected. However,

their resolution is limited by the size of an atmospheric cell. The resolution, s ¼ =r0,so the effective aperture of the telescope is equivalent to the size of the atmospheric

cell, not the telescope mirror. If r0 ¼ 10 cm, then a 10 cm diameter telescope will have

the same resolution as a 10 m telescope (100, Eq. 2.3). For the 10 cm diameter or

smaller telescope, the resolution is diffraction-limited and for any larger telescope, it

(a)

Winddirection

Winddirection

(b)

(c)

Figure 2.14. Effect of the atmosphere on a stellar image. (a) If the telescope were in space, theimage of a point source would be the diffraction pattern of the telescope with the width ofthe central peak given by d. (b) Individual atmospheric cells of typical size, 10 cm, move acrossthe telescope aperture effectively breaking up the aperture into many small apertures about thissize. Since the refraction is slightly different for the different cells, each one creates its own image,called a speckle, on the detector at a slightly shifted location. Each individual image has its owndiffraction pattern and the image changes rapidly with time as the atmospheric cells move acrossthe aperture. (c) If a time exposure of the star is taken, the speckle pattern is seen as a blendedspot, called the seeing disk.


will be seeing-limited. Depending on the linear size of the pixels on the CCD, however,

the resolution could also be pixel-limited (Example 2.1). The image of a point source

on the detector, as actually measured, is called the point spread function (PSF).

Example 2.1

An 8 inch f/10 telescope is outfitted with a 1024 1024 pixel2 CCD, each pixel 9 m square.Determine the field of view, and also the resolution when this telescope is used, (a) in aback yard with seeing of s ¼ 200 and (b) on a high mountain with s ¼ 0:400. (c) Could thistelescope achieve higher resolution at either of these locations?

Using cgs units, the telescope diameter, D ¼ 20:32 cm and pixel size

lpix ¼ 9 104 cm. The CCD size is ldet ¼ 1024 lpix ¼ 0:92 cm and the focal length,

f ¼ 10 D ¼ 203:2 cm. From Eq. (2.1), the field of view is FOV ¼ ldet=f ¼4:5 103 rad, or 15:60 square, independent of location. By the same equation, the

pixel field of view (converting to arcseconds) is pix ¼ 0:9100. The diffraction limit of

the telescope, from Eq. (2.3), adopting an observing wavelength of ¼ 507 107 cm

(Table G.5), is d ¼ =D ¼ 2:5 106 rad ¼ 0:500.

(a) Since s is larger than both the pixel resolution, pix, and the diffraction-limited

resolution, d, the observations are seeing-limited and the resolution is 20010.

(b) In this case, pix > d > s so the observations are pixel-limited and the resolu-

tion is 0.9100.

(c) No improvement can be achieved in the backyard case since the seeing cannot be

changed. However, on the mountain, by changing the CCD to one which has pixels

about half the size or smaller, the telescope could become diffraction-limited with a

resolution of 0.500.

2.3.3 Adaptive optics

One solution to the problem of seeing is to launch telescopes, even those working at

optical wavelengths, into orbit as was done, for example, for the Hubble Space

Telescope. This solution, however, is very expensive. Another is to improve the design

of observatories to minimise local temperature gradients that can produce turbulence

(Figure 2.15).

However, there is now a very effective way of dealing with the problem of seeing –

that of adaptive optics. Now being routinely employed at major facilities in the near-IR

where the atmosphere is more stable, adaptive optics is a method of correcting for the

10A more accurate treatment would consider how each effect contributes to the PSF.


seeing in real-time bymaking rapid adjustments to a deformable mirror. An example of

an adaptive optics system is shown in Figure 2.16.

After passing through the telescope aperture, the signal, along with a reference

signal from a point source, is reflected from a deformable mirror. The reference signal

is then split off to a ‘wavefront sensor’ which must sample the wavefront on ms

timescales. The wavefront sensor breaks up the reference signal into many images via

Figure 2.15. The Gemini North Telescope on Mauna Kea, in Hawaii, showing the open sides whichhelp to minimise local turbulence. (Reproduced by permission of Neelan Crawford, courtesy ofGemini Observatory/AURA)

Figure 2.16. Simplified diagram of an adaptive optics system. Adapted from http://cfao.ucolick.org/ao/how.php)


lenses. If the wavefront is planar, then the reference signal images are evenly spaced on

the detector, but if the wavefront is distorted, then the images are unevenly spaced.

After analysis of the spacing, a signal is sent to actuators at the rear of the deformable

mirror which adjust the shape of the mirror to compensate for the wavefront distortion.

Since the signal of astronomical interest is also reflecting from the same deformable

mirror, its image is therefore corrected as well.

In order for this system towork, the reference signal must be bright andmust be close

enough in the sky (<10) to the astronomical source of interest that the two signals pass

through the same atmosphere and same optical path. However, there are very few bright

stars in the sky that are fortuitously placed close to any arbitrary astronomical source.

The solution has been to generate a laser guide star. A laser beam is emitted into the sky

and excites or scatters off of particles in the atmosphere at an altitude of about 100 km,

forming a point-like source as the reference beacon. A natural guide star must still be

used for absolute positional information, but it need not be so bright or close to the

target source.

The image quality that has been achieved by adaptive optics is remarkable

(Figure 2.17) and this technique promises to provide more improvements as the

technology becomes more refined. At present, adaptive optics systems have been

Figure 2.17. Example of the improvement that can result using adaptive optics. The image at leftshows many stars crowded together in the region of the Galactic Centre. The image appears out offocus because of the seeing. At right is the same field after using adaptive optics. The star imagesnow show the diffraction pattern of the telescope and the resolution approaches the diffractionlimit. (Reproduced by permission of the Canada-France-Hawaii Telescope/1996)


added on to existing telescopes, but they will be an integral part of the design of new

and larger telescopes in the future (Ref. [65]).

2.3.4 Scintillation

Scintillation, commonly called ‘twinkling’, is the rapid change in amplitude (bright-

ness) of a signal with time due to the atmosphere. It occurs for similar reasons as

seeing, as atmospheric cells act like lenses focussing, defocussing, and shifting the

signal. Scintillation, however, is primarily due to changes in the curvature of the

wavefront rather than its tilt and there is a greater effect from atmospheric cells that

are higher up, rather than near the ground. The most characteristic timescale for

scintillation is of order a ms, but variations are seen on a range of timescales from

microseconds to seconds or longer (Ref. [53]). The fluctuations in amplitude will be

about a mean value which is the desired measurement and, since most exposure times

are longer than the scintillation timescale, scintillation is not a serious problem for

most astronomical measurements.

A star, or other point-like source, viewed through the atmosphere can be seen by eye

to fluctuate in intensity. An extended object has a brightness distribution that is

convolved with the seeing disk. That is, the object’s brightness distribution can be

thought of as a series of point sources of different brightness, each one ‘blurred’ to the

size of the seeing disk and each seeing disk is scintillating. If the eye could spatially

resolve each of these points, it would see brightness fluctuations across a source.

However, the resolution of the human eye ( 10) is much poorer than the seeing (100).Thus, each of the fluctuations across an extended source (some positive, some negative

about the mean) are not seen individually by the eye but are instead integrated together.

The result is that the eye perceives an extended source as steadily shining. The old

adage ‘stars twinkle and planets shine’ is true for the planets visible by eye, but

exceptions can occur (Prob. 2.9).

2.3.5 Atmospheric reddening

Reddening is the result of wavelength dependent absorption and scattering. Shorter

wavelength (blue) light is more readily scattered out of the line of sight than is light of

longer (redder) wavelengths. Therefore longer, redder wavelengths will have greater

intensity when viewed through an absorbing and scattering medium like the atmo-

sphere. This is true at both the zenith and the horizon, but since viewing towards the

horizon corresponds to viewing over a longer path length, the effect is stronger, as is

readily seen when viewing a red sunset. Reddening of starlight can also be noticed by

eye by observing a given star when it is high in the sky and then again when the star is

about to set. Molecules as well as dust and aerosols contribute to the reddening in the

Earth’s atmosphere. Absorption and scattering will be treated in greater detail in

Chapter 5.


2.4 Processing the signal

It is not possible to detect a signal without affecting it. Both the atmosphere, as well as

the telescope and detector, impose their spatial and spectral response patterns on the

signal, as already seen. As light passes through other optical devices within the

telescope (for example, through lenses, wavelength-dispersive media such as spectro-

graphs, through filters or reflected from smaller mirrors), each ‘interaction’ also affects

the signal. Moreover, other sources of emission which can pollute the image (such as

detector noise or atmospheric emission) must be removed. The image that is detected,

called the raw image, requires corrections for these effects, or at least a characterisation

of them, to know their limitations on the data. In addition, the corrected data, which

consist of an array or arrays of numbers, must be converted to commonly-understood

units (e.g. magnitudes or Jy per unit solid angle) that are independent of telescope or

location, i.e. the signal must be calibrated. The end result will be a processed image

such as would be measured with ‘perfect’ instrumentation above the Earth’s atmo-

sphere.

2.4.1 Correcting the signal

The steps involved in correcting the signal vary depending on waveband, the specifics

of the telescope and its instrumentation. For example, if the telescope is in orbit,

corrections are required for variations in the signal due to variations in the orbit.

If images are taken rapidly, a new image may retain a ‘memory’, due to incomplete

CCD readout from the previous image. Data from radio telescopes require radio

interference to be excised and beam sidelobes to be carefully characterised. Optical

telescopes must have unwanted cosmic ray hits removed and atmospheric emission

lines either corrected for or avoided. Non-uniform responses across the field of view as

well as in frequency must be ‘flattened’ and higher order distortions corrected if

present.

For work with an optical telescope and CCD, the minimum corrections would

include the following steps. The CCD image would first have a constant positive offset

subtracted from the image to account for a known DC voltage level that exists in CCDs

(the bias correction). The signal that results from thermal electrons in the detector,

without any exposure to light would then be subtracted (the dark or thermal correction).

The non-uniform response of each pixel to light across the CCDmust be corrected; this

is done by dividing the image by another image that has been exposed to a source that is

known to be evenly illuminated (the flat field correction). Multiple frames of the same

source, if they are present, must be medianed (rather than averaged) together, a process

which removes random cosmic rays (Sect. 3.6). Finally, a background sky emission

level, determined from a region on the CCD in which there is no source, would be

subtracted from the image. Depending on the complexity of the instrumentation, it is

usually more time-consuming to correct and characterise the data than to acquire the

data in the first place.

2.4 PROCESSING THE SIGNAL 49

2.4.2 Calibrating the signal

The most common method of calibrating the signal is to compare it to a reference

signal (the calibrator) of known flux density or brightness. There are usually only a few

primary flux calibrators at any wavelength and all other observations are brought in

line with them. The stars, Vega and Sirius, are good examples in the optical band, as are

the quasars (see Sect. 1.6.4), 3C 286 and 3C 48 at radio wavelengths. Asteroids and

planets are often used at sub-mm and/or IR wavelengths. Secondary calibrators may

also be used. These would be sources whose fluxes are tied to those of the primary

calibrators and are more widely dispersed in the sky so that one or more are accessible

during any observing session.

Atmospheric attenuation is important at optical and sub-mm wavelengths and

increases with zenith angle. At optical wavelengths, a calibrator must be monitored

periodically at a variety of zenith angles over which the source of interest is also

observed. The source data are then corrected according to the calibrator signal at the

corresponding zenith angle. At sub-mm wavelengths, at least once per night a measure-

ment of sky brightness as a function of zenith angle is made (called sky dips) and this

information is combined with a model of the atmosphere to compute the attenuation.

Once the calibrators have been observed, it is then a straightforward matter of

multiplying the source counts, in arbitrary units, by the factor determined from the

calibrator (e.g. magnitudes/counts, Jy/counts, etc.). Since the calibrator and source are

observed with the same equipment, the attenuation of the signal due to absorptions and

scatterings in the instrumentation are also taken into account.

Clearly, the primary flux calibrators must be measured on an absolute scale for this

system to work. This process is non-trivial and considerable effort has been expended

to make accurate measurements and bring them to a common system (e.g. Refs. [11],

[136]). It consists of observing the source along with a man-made source of known

brightness. At radio or sub-mm wavelengths, this might be a black absorbing vane (a

black body, see Sect. 4.1 and Example 4.1) at fixed known temperature and, at optical

wavelengths, a calibrated lamp. The flux density of Sirius and Vega are also alterna-

tively determined by calculating the expected value, using other known stellar para-

meters such as temperature, distance, and the shape of the spectrum. Absolute

calibrations are also sometimes used during normal observations, rather than reverting

to the relative scale discussed above, particularly at radio and sub-mm wavelengths.

2.5 Analysing the signal

The final result is a reduced, or processed image, in units that are common to astronomy,

ready for analysis, measurement, and other scientific scrutiny. Measurements of specific

intensity and flux density can be made as described in Sect. 1.3 and illustrated in

Figure 1.8, and other parameters can then be determined, if possible, as described in the

previous chapter. It is therefore important to understand the characteristics of the images

in order to place limits on the source parameters which are derived from them.


Once an image has been obtained (assuming no artifacts), it will be characterised by

eight properties, three providing spatial information, three providing spectral informa-

tion, and two related to the amplitude of the signal:

(1) Position of map centre: The map centre is usually placed at the location of the

target source in the sky and specified by some coordinates, Right Ascension and

Declination (RA, DEC) being most common.

(2) Field of view: The FOV is set by the detector size in the case of a simple system

(Sect. 2.2.3) though it can be smaller for more complex, partially blocked light paths.

(3) Spatial resolution: This is affected by telescope size, pixel size, or seeing, as

described in Sect. 2.3.2.

(4) Central wavelength or frequency: Different instrumentation is required for

different wavebands but, within any given band, more accurate centering in wavelength

or frequency can be achieved by tuning or using filters.

(5) Bandwidth: Implicit in the setting of central wavelength or frequency is the fact

that a range of wavelength (1 to 2) or frequency (1 to 2) will be detected.

(6) Spectral resolution: If detailed spectral information is required, the waveband

may be broken up into wavelength or frequency channels and each examined

separately.

(7) Noise level: A spatial region of the image that is devoid of emission (after all

corrections and sky subtraction) will contain positive and negative values about a mean

of zero due to noise in the system. The value is measured as a root-mean-square, i.e.

N ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi< x2 >

p, where x is the calibrated pixel value and the angle brackets represent an

average.

(8) Signal-to-noise ratio (S/N): This is the ratio of the signal strength at any image

location to the rms noise. In general, longer integration times (that is, longer exposure

times) result in higher S/N images because N decreases with increasing integration

time. In order for a signal to be considered a real detection, the S/N ratio must be high

enough to be sure that it is not just a random peak in the noise level. A minimum value

of 3 to 10 will often be used, although other factors can also play a part in this

decision, such as the angular size of the signal on the detector or whether it is seen in the

same location at other wavebands. Another way of characterising the image as a whole

is by the dynamic rangewhich is the ratio of the peak signal to the minimum detectable

signal on the map. This is therefore a measure of the ‘stretch’ in brightness that has

been detected. A practical measurement is to take the peak signal on the map over the

rms noise. The S/N ratio may be different at every point in the map but there is only one

value of dynamic range for a map.

2.5 ANALYSING THE SIGNAL 51

These properties must be kept in mind when determining and interpreting the

scientific results that derive from the data. Example 2.2 provides an illustration of

the kinds of questions one should ask prior to embarking on an observing run.

Example 2.2

A galaxy at a distance of D ¼ 25 Mpc will be observed with a spatial resolution of 1’’ at afrequency of 4:57 1014 Hz and bandwidth of ¼ 2 1012 Hz in order to detect theHII regions (these are ionised hydrogen regions around hot stars, see Figures 3.13 and 8.4)within it. The map noise expected for the integration time proposed is N ¼ 21017 erg s1 cm2 Hz1 sr1. Could a spherical HII region of typical diameter, d ¼ 500 pc,and luminosity, L ¼ 1039 erg s1, in this frequency band be detected from these observations?Would it be resolved?

AtD ¼ 25Mpc ¼ 7:71 1025 cm, the flux (Eq. 1.9) is f ¼ 1:34 1014 erg s1 cm2 and

the mean flux density in a band of width, 2 1012 Hz, is (Eq. 1.7) f ¼ 6:691027 erg s1 cm2 Hz1. A diameter of 500 pc corresponds (Eqs. B.1, B.3) to a solid

angle of ¼ 3:14 1010 sr so the specific intensity is (Eq. 1.13) I ¼ 2:131017 erg s1 cm2 Hz1 sr1. This is just at the rms noise level, so the HII region would

not be at the minimum level for detection. A longer integration time should be chosen to

increase the S/N. An angular resolution of 100 corresponds (Eq. B.1) to a linear scale of 121 pcat the distance of this galaxy, so the HII region would indeed be resolved, if it were detected.

2.6 Visualising the signal

The final image can be displayed in a variety of ways. The galaxy, NGC 2903, has

already been shown for different resolutions (Figure 2.11). Figure 2.18 shows the same

galaxy, but using three different visualisation schemes. In all three cases, the data (i.e. the

numerical values in the 2-D array) are exactly the same. The first image shows the data as

a linear greyscale (Figure 2.18.a) similar to the linear response of the CCDwith which it

was obtained. This represents the data well, except that the range of brightness is very

large and it is difficult to show the entire range on paper. Details near the centre of the

galaxy are ‘burned out’ and the faint outer regions of the galaxy do not show upwell. An

alternative is Figure 2.18.b which applies a logarithmic weighting to the data prior to

displaying it. This has the effect of compressing the range of brightnesses so that both the

bright and dim parts of the image are easily seen. The drawback is that there is less

contrast in any part of the image so the true range of brightness is not obvious. Figure

2.18.c is a third possibility. Apart from being visually appealing, the use of colour,

depending on how it is applied, allows details in both the bright and dim parts of the

image to be highlighted. This is called false colour because the adopted colour scheme is

chosen only for illustrative purposes and has no scientific meaning otherwise.

Figure 2.19 illustrates a more sophisticated representation of data when comparisons

are desired between wavebands. Contours (Figure 2.19.a) are most often used alone or


Figure 2.18. An optical image of the galaxy, NGC 2903 (distance, D ¼ 8:6 Mpc), taken in a bandcentred at 500 nm, is represented in three different ways: (a) linear greyscale, (b) logarithmicgreyscale, and (c) logarithmic false colour. (see colour plate)

Figure 2.19. (a) Radio continuum contours at 20 cm overlaid (single-channel bandwidth ¼0.66 MHz) on the same image of NGC 2903 as in Figure 2.18. The radio contours are at 0.6, 1.5, 3, 5,10, 25, and 50 mJy beam1, the peak map level is 117 mJy beam1, the beam FWHM is 54.400 (seeSect. 2.2.4), there are 1.700 per pixel, and the rms noise is 0.5 mJy beam1. (b) Combination ofthree different images in three wavebands centred at 500 nm (shown as blue), 650 nm (green)and 820 nm (red). (see colour plate)

2.6 VISUALISING THE SIGNAL 53

in overlays like this when the spatial resolution is low or the image structure is not very

complex. Otherwise the contours become crowded. In Figure 2.19.b, another optical

image is shown. At first glance, this might be interpreted as being a single image for

which colours have been applied (i.e. false colour). In fact, it is a combination of three

different images taken in three different wavebands for which different colours have

been adopted. Again, the colours corresponding to these wavebands were not intended

to describe what the eye would see. They are simply chosen to distinguish between the

three bands in order to convey more information. For scientific analysis, it is best to

consider the image in each band separately.

Any of these approaches is valid and astronomers generally use whatever weighting,

contour, or colour scheme allows them to convey the information about the source that

is desired. Since the ‘true’ colour of an astronomical object applies only to the narrow

visual range of the electromagnetic spectrum to which the eye is sensitive, from a

scientific point of view there is little motivation to reproduce true colour images of

astronomical objects. However, the careful adoption of a visualisation scheme can

sometimes provide useful information ‘at a glance’, as Example 2.3 illustrates.

Example 2.3

Consider the strong background source just to the north of the HII region, IC 5146, that isvisible in the diffraction-limited radio image, shown by contours, in Figure 8.4.

(a) What quantity is represented by the contours?

(b) Is this source resolved or unresolved?

(c) Estimate the flux density of this source (Jy).

(d) Express the peak specific intensity of this source in cgs units.

(a) The contour units are in milli-Janskys (see Eq. 1.8) per beam-solid-angle, (mJy

beam1) so these are contours of specific intensity.

(b) If the source is unresolved (source angular diameter, S; << the beam angular

diameter, b), then its observed size will be the same as the beam size since its emission

is spread out over the shape of the beam (Sect. 2.2.4). If S approaches the size of the

beam or exceeds it, then the observed size of the source will be broader than the beam.

The source in the figure is the brightest source in themap (see contours) so its peak is (see

caption) Imax ¼ 318 mJy beam1. The half-maximum contour is then I ¼ 159 mJy

beam1. Measuring the full width of the half-maximum contour using the tick marks on

the Declination (ordinate, or north-south) axis11 shows that the apparent source size,

11Do not use the Right Ascension axis to measure angles since it is expressed in units of time.


though somewhat oval in shape, is not significantly greater than the stated beam

FWHM of 8000 in any direction. Therefore this source is unresolved, a conclusion

that is reached by a quick comparison of its angular size at half-maximum with the

beam size.

(c) The flux density, fS, of an unresolved source is preserved (Figure 1.9,

Sect. 2.2.4). Since the main beam has a Gaussian shape and the telescope is pointing

directly at the source, the integral under the beam (of solid angle, b) can be

approximated as (Eq. 1.13), f ¼RI cosðÞ d Imaxb ¼ 318mJy. Basically,

the flux density of an unresolved source is confined to a single beam, and therefore

read simply from the peak at its position for contours in ‘per beam’ units.

(d) This part requires more than a ‘quick glance’ at the map. From the definition of a

milli-Jansky (Eq. 1.8), I ¼ 318 1026 erg s1 cm2 Hz1 beam1. Converting the

beam FWHM to radians, we find, b ¼ ð80=60=60=180Þ ¼ 3:88 104 rad. Then

the beam solid angle is (Eq. B.3 for a beam that is approximately circular on the sky),

b ¼ b2=4 ¼ 1:18 107 sr. We finally find, I ¼ ð318 1026 erg s1 cm2

Hz1 beam1Þ ð1:18 107 sr beam1Þ ¼ 2:7 1017 erg s1 cm2 Hz1 sr1 in

cgs units.

There is one possible element of an image that is still missing, however – that of the

spectral dependence of the emission. The above images apply only to data obtained in a

single spectral channel. However, if the frequency band has been split into many

frequency channels (or equivalent in thewavelength regime), then an image is obtained

in each of these channels. Rather than a two-dimensional image, we now have a three-

dimensional data cube, the third axis being either frequency, or wavelength. If emis-

sion is observed in a spectral line (see Chapter 9), then the Doppler relation (Table I.1,

Sect. 7.2.1) can be used to convert the third axis into a velocity. Figure 2.20 illustrates

this for neutral hydrogen (HI, see Sect. 3.4 for nomenclature) data obtained for NGC

2903. Data cubes like this contain a rich wealth of information about the source, its

contents and dynamics.

Problems

2.1 Using Figure 2.4 as a starting point, (a) draw a diagram showing the rays that indicate

the field of view on the sky, and (b) draw a diagram which illustrates why larger apertures

should result in brighter images.

2.2 For the conditions given in the Appendix at the end of this chapter, determine the

ellipticity (ratio of minor to major axis) of the sun when its lower limb appears to be at the

horizon.


2.3 (a) An observer, using the V-band filter (see Table 1.1), centres a star on his 256 256

pixel CCD detector. The star’s observed zenith angle is 60, the CCD pixel separation

corresponds to 100 on the sky, and the seeing is sub-arcsecond. He then changes to a U-bandfilter followed by an I-band filter. Determine and sketch where the star will appear on the

CCD in each of the filters, as well as its true position for the reference pressure and

temperature given in the Appendix at the end of this chapter.

(b) Repeat part (a) but for a temperature of 0C.

(c) If the filters are removed and if the CCD could detect all wavelengths from the U

band to the I band with uniform sensitivity, what would the image of the star look like on

the CCD?

2.4 For observations at 2.2m through atmospheric turbulence,

(a) Derive an expression for the variation in the index of refraction, n, with tempera-

ture, T. For a single layer within the turbulent region, write the variation in the refractive

angle, R, as a function of T (see Eq. 2.A.2).

Figure 2.20. A data cube showing 21 cm HI emission from the galaxy, NGC 2903 in a three-dimensional ‘space’ with the x axis being Right Ascension, the y axis being Declination, and thez axis being either wavelength, frequency, or velocity. For this galaxy, the emission on the z axisextends from an observed wavelength of 1 ¼ 21:13215 cm to 2 ¼ 21:15837 cm, with the centreat sys ¼ 21:14526 cm. An analysis of the emission in this cube can determine how the galaxy isrotating and how much mass is present (Sect. 7.2.1).


(b) Evaluate R (arcseconds) for a star at an observed zenith angle of 45 when the

fluctuation in temperature, T ¼ 0:3K, and for typical high altitude conditions, i.e.

P ¼ 600 102 Pa; T ¼ 0C.

(c) Assuming that the combination of different R from different layers along the line of

sight results in a seeing size of s 10 R, determine the corresponding atmospheric cell

size, r0.

(d) For a wind speed of 20 km/h, how frequently must the deformable mirror be adjusted

in an adaptive optics system to correct for seeing at this wavelength?

2.5 Determine whether the dark-adapted human eye is diffraction limited, seeing limited,

or pixel-limited (Table G.5).

2.6 (a) Determine the spatial resolutions (arcsec) of the following instruments (assume

that seeing for the ground-based instruments is 0.500): (i) The James Webb Space Telescope

(JWST, D ¼ 6:5m; Deff ¼ 5:6m) at 28m. (ii) The James Clerk Maxwell Telescope

(JCMT) (D ¼ 15m) using the 2nd Submillimetre Common-User Bolometer Array

(SCUBA-2) detector (Deff ¼ 12:4m) at 450m.

(b) How largewould a ground-based optical telescope operating at 507 nm have to be to

match the resolution of the JWST? How largewould a single dish radio telescope operating

at 20 cm have to be?

2.7 Using the information in the caption of Figure 1.2, determine the proper motion

(in arcsec/year) of the filaments in Cas A. How long would it take for this proper

motion to become measurable, using a radio telescope whose effective diameter is,

Deff ¼ 32 km?

2.8 A galaxy is face-on, circular in appearance, and subtends an angular diameter of 1000.In which of the following situations would the galaxy be like a point source, i.e. completely

unresolved: (a) an amateur ‘backyard’ 2 inch diameter optical telescope with seeing of 200,(b) the largest single-dish radio telescope in the world (Arecibo, Puerto Rico) of diameter,

305 m, operating at 20 cm, (c) the 0.85 m Spitzer Space Telescope (launched 2003)

operating at the IR wavelength of 60m?

2.9 On a dark clear night but with poor seeing (s ¼ 400), the two planets, Jupiter and

Uranus (both near opposition) are viewed by eye. Determine, approximately, the number of

independent seeing disks that are present across the face of these two planets and indicate

whether they will twinkle or shine.

2.10 From the image and information provided for the radio continuum data in

Figure 2.19.a,

(a) list or estimate numerical values of the eight map parameters, as described in

Sect. 2.5.

(b) estimate the following values (cgs units) for the galaxy in the 20 cm band : (i) the

flux density, (ii) the flux, (iii) the luminosity, (iv) the spectral power.


2.11 A radio telescope operating at 6 cm has a circular beam of FWHM ¼ 3000. If thelowest possible rms noise on a map made using this telescope is 0:01mJy beam1, could

this telescope detect radio emission from the hot intracluster gas whose spectrum is plotted

in Figure 8.3?

Appendix: refraction in the Earth’s atmosphere

Consider a star whose light impinges on the Earth’s atmosphere at an angle, 1, fromthe vertical (Figure 2.A.1). In the simplest approximation, one could consider a ray to

pass from the vacuum of space, with an index of refraction, n1 ¼ 1, to an atmosphere

which is a uniform slab with index of refraction, n2. An observer at position, O, would

then see the star at a zenith angle of 2 in the sky. Thus, the star appears higher in the skythan it actually is. The two angles are related via Snell’s Law (Table I.1),

sinð2Þ ¼1

n2sinð1Þ ð2:A:1Þ

The Earth’s atmosphere is not uniform, but it is still possible to consider horizontal

slabs of atmosphere which are thin enough that, within each, the density is uniform and

the index of refraction is constant. The geometry of the refraction in each thin slab is

then described by the figure and Eq. (2.A.1) now applies only to the first slab. The slab

immediately underneath the first slab would then have an incoming angle of 2 and anoutgoing angle, 3. Thus,

sinð3Þ ¼n2

n3sinð2Þ

θ2

θ2

n2

n1

θ1

O

Figure 2.A.1. A ray diagram, showing the direction of the incoming wave front and its refractionin an atmospheric layer (angles exaggerated).


From Eq. (2.A.1), this becomes,

sinð3Þ ¼1

n3sinð1Þ

Thus, for an arbitrary number of slabs, N,

sinðNÞ ¼1

nNsinð1Þ

We now identify N with the observed zenith angle, zo, 1 with the true zenith angle,zt, and nN with the index of refraction at ground level, n, to write,

sinðzoÞ ¼1

nsinðztÞ

Expressing zt ¼ zo þ R, where R is the refractive angle, using the identity,

sinðzo þ RÞ ¼ sinðzoÞ cosðRÞ þ cosðzoÞ sinðRÞ and assuming that the angle, R, is small

so that sinðRÞ R and cosðRÞ 1, we find, for the plane parallel atmosphere,

R ðn 1Þ tanðzoÞ ð2:A:2Þ

Eq. (2.A.2) gives the change in zenith angle (radians) as a function of observed zenith

angle and depends on the index of refraction as measured at the position of the

observer. The source always appears higher in the sky (smaller zenith angle) than its

true position.

A more accurate derivation (Ref. [163]) results in,

R ¼ A tanðzoÞ B tan3ðzoÞA ¼ ðn 1ÞB ¼ 0:001254ðn 1Þ 0:5ðn 1Þ2

ð2:A:3Þ

where R, A and B are in radians. Eq. (2.A.3) is valid12 for zenith angles zo < 75.The index of refraction, n, is a function of wavelength, , and also depends on the

temperature, T, pressure, P, and water vapour content of the air. An expression which is

valid from the 230 to 1690 nm (ultraviolet through infrared) is, for dry air13,

ðn 1Þ ¼ 5:792105 102

238:0185 12

þ 1:67917 103

57:362 12

" #

P

Ps

Ts

T

ð2:A:4Þ

12The variation in R due to varying atmospheric scale height with temperature as well as the non-sphericity of theEarth are less than 1 per cent and have therefore been neglected in the expressions for A and B.1

APPENDIX: REFRACTION IN THE EARTH’S ATMOSPHERE 59

for T in K, P in Pa, and in m (Ref. [41], [176]). The reference temperature and

pressure are Ts ¼ 288:15K, and Ps ¼ 1013:25 102 Pa, respectively.

The resulting constants, A and B, are plotted as a function of wavelength in

Figure 2.A.2. The resulting refractive angle is plotted against observed zenith angle

for a wavelength of 574 nm in Figure 2.A.3.a.

57

58

59

60

61

62

63

0.4 0.6 0.8 1 1.2 1.4 1.6 0.063

0.064

0.065

0.066

0.067

0.068

0.069

0.07

0.4 0.6 0.8 1 1.2 1.4 1.6

(a) (b) C

on

stan

ts o

f R

efra

ctio

n (

arcs

ec)

Wavelength (micron)

Figure 2.A.2. Plot of the constants, (a) A and (b) B, for a temperature and pressure of 288.15 Kand 1013:25 102 Pa, respectively.

0

50

100

150

200

0 10 20 30 40 50 60 70200

400

600

800

1000

1200

1400

1600

2000

1800

74 76 78 80 82 84 86 88 90

(a) (b)

Ref

ract

ive

An

gle

(ar

csec

)

Observed zenith angle (degrees)

Figure 2.A.3. The refractive angle, R, as a function of observed zenith angle at a wavelength ofl574 nm, for dry air at T ¼ 288:15 K and P ¼ 1013:25 102 Pa. (a) Values for zenith angles up to75 evaluated from Eq. (2.A.3.) (b) Values for large zenith angles from Ref. [176]. Depending onatmospheric conditions, there can be significant variations in this plot.


For zenith angles that are large (zo Z 75) the index of refraction becomes more

strongly dependent on local atmospheric conditions and modifications must be made to

the equations presented here. For example, at sea level, a temperature variation of 65Ccan result in a variation in refraction at the horizon by as much as 150 (Ref. [176]).Corrections for variations in pressure and water vapour content are typically smaller

than this, but can still be significant. An example of the refraction at large zenith angle

is shown in Fig. 2.A.3.b and the values are given in Table 2.A.1.

Table 2.A.1. Refractive angle for large zenith anglesa

zoðdegÞ R (arcsec) zo (deg) R (arcsec)

74 196.49 84 497.25

76 225.00 85 578.72

78 262.20 86 688.25

80 312.78 87 841.19

81 345.52 88 1064.59

82 385.34 89 1408.82

83 434.68 90 1974.35

aFrom Ref. [176], for dry air at sea level and 45 latitude,T ¼ 288:15K; P ¼ 1013:25 Pa, and 574 nm.

APPENDIX: REFRACTION IN THE EARTH’S ATMOSPHERE 61

PART IIMatter and Radiation Essentials

Radiation and matter can be thought of as different manifestations of energy. Consider,

for example, E ¼ h n or E ¼ m c2 (Table I.1). These different forms of energy, how-

ever, interact with each other in very specific ways. Were it not for this fact, we could

discern very little about the Universe around us. In astrophysics, it is radiation that we

observe, but it is matter that we most often seek to understand. What is its origin? How

does it behave?What is its energy source? By what process does it radiate? How does it

evolve with time? Is the radiation that we see even a good indicator of the true content

of the Universe?

These are challenging questions and not easily answered. We are not left without

some tools, however. The radiation that we detect does indeed provide us with evidence

as to the nature of this vast home in which we live. Sometimes, the evidence is strong

from a single set of observations that are thorough and well executed. Sometimes it is

strong because it comes from completely independent data sets which, when consid-

ered together, present a consistent and compelling picture. Sometimes we only have

clues and hints as to the direction that the truth lies. Science is a process, not an end-

point, providing us with a snapshot of our natural world. We would always like that

snapshot to be brighter, clearer, more detailed or larger, but to have a picture is

infinitely better than being left in the dark. Even more encouraging is the fact that

we continue to see the picture focussing and refocussing before our eyes.

To understand this process, or to be a part of it, there are some important concepts

related to matter and radiation that should first be appreciated. In this section, therefore,

we present some ‘essentials’ of matter and radiation in preparation for a discussion of

the interaction between these two forms of energy. It is this matter/radiation interaction

which is at the heart and soul of astrophysics.

3Matter Essentials

3.1 The Big Bang

The Universe, including space, time, energy in its various forms, virtual particles and

photons, visible and unseen matter, is believed to have begun in a primordial fireball

called, prosaically, the Big Bang. It is important to think of the Big Bang, not as an

event that has occurred at some point in time and whose result, the Universe, is now

expanding into space. Space and time, as we know them, did not pre-exist, but rather

came into being with the Big Bang. There is now much evidence for this event that,

collectively, presents a compelling case. For example, a variety of observations suggest

that the Universe has evolved. Since light travels at a finite speed, as we look farther

into space, we also look back to earlier epochs in the history of the Universe. This early

Universe was very different from what we see today, showing different kinds and

admixtures of sources, and objects with properties that differ from those that are local

(Figure 3.1). The expansion of the Universe itself (Appendix F) is consistent with this

view and the age of the Universe, derived from its cosmological properties (13.7 Gyr,

Ref. [16]), is similar to the age of the oldest stars (12.7 to 13.2 Gyr, Ref. [73])1. The fact

that the night sky is dark is also consistent with a Universe of finite age (Sect. 1.4). At

the earliest epoch that it is currently possible to observe, we see the cosmic microwave

background (CMB), an opaque glow at 2.7 K that fills the sky in all directions

(Figure 3.2). The CMB is an imprint of the cooling primordial fireball. Tiny fluctua-

tions in the CMB are now revealing further details about the cosmological parameters

that characterize the large-scale properties of the Universe. These fluctuations form the

seeds from which galaxies later grow (see Figure 5.7). Finally, the abundances of the

light elements, as predicted from nucleosynthesis in the first moments after the Big


1The error bar on the age of the Universe is claimed to be less than 1 per cent and on the stellar ages, it is likely lessthan 5 per cent.

Bang, are in agreement with the observed values. Any theory that seeks to supplant this

model would have to do a better job of addressing each of these points.

3.2 Dark and light matter

What of the matter produced in the Big Bang? Strangely, most of it is invisible to us.

The bulk of the material created in the immediate aftermath of the Big Bang is known

as dark matter because, to the limits of the available data, this material does not emit

light at any wavelength. Its presence is inferred by the effect it has on neighbouring

matter that can be seen. For example, galaxies rotate much faster in their outer regions

than would be expected from the amount of mass that is inferred from the visible

galaxy. This suggests that there is dark matter surrounding galaxies in a distribution

that is more extended and more spherical than the visible material (see Figure 3.3).

Similar arguments apply to clusters of galaxies. Adding up the mass from each member

galaxy as well as any intracluster gas (e.g Figure 8.8) results in a value that is far too

low to bind the cluster together. These clusters are believed to be bound because the

time it would take for the gas to escape and the galaxies to disperse is quite small if they

Figure 3.1. The Hubble Ultra-Deep Field, taken from the Hubble Space Telescope in a 106 sexposure. The red ‘smudges’, circled in these images (several blow-ups shown on right) are amongthe most distant galaxies ever observed optically. The faintest objects are 2:5 1010 times asbright as what can be observed with the naked eye. These galaxies, or rather ‘galaxy precursors’, areseen as they were at a time when the Universe was only 5 per cent of its current age. They are muchsmaller than the giant systems we see today and suggest that today’s galaxies are formed by smallersystems building up into larger ones. (Reproduced by permission of NASA, ESA, R. Windhorst(Arizona State University), and H. Yan (Spitzer Science Centre, Caltech)) (See colour plate)

66 CH3 MATTER ESSENTIALS

Figure 3.2. Temperature map of the Cosmic Microwave Background (CMB) radiation as measuredby the Wilkinson Microwave Anisotropy Probe (WMAP) satellite, launched in 2001. This is aHammer–Aitoff projection in which the whole sky is shown in an elliptical projection. The CMB isremnant black body radiation (see Sect. 4.1) from a time 380 000 yr after the Big Bang when thetemperature of the Universe cooled to about 3000 K, sufficient for electrons and protons to combineand form neutral atomic hydrogen. At this time, corresponding to a redshift of z ¼ 1000 (see Sect.7.2.2), the Universe went from being opaque to being transparent. The mean observed temperatureof the CMB, as we now measure it, is 2.725 K, but tiny fluctuations of temperature are shown in thisimage of order þ200mK (light) to 200 mK (dark) (Ref. [16]). The temperature fluctuations arecaused by acoustic, or sound waves. The angular scales over which perturbations can be seenprovide information that constrains the cosmological parameters of our Universe. (Reproducedcourtesy of The WMAP team and NASA)

Figure 3.3. Sketch of the Milky Way galaxy face-on (left) and edge-on (right). The Milky Way is abarred spiral galaxy and the Sun is located 8 kpc from the nucleus within the disk. The disk is thinin comparison to its diameter, only about 300 pc thick. It is an active region containing stars, gasand dust, and star formation occurs within it. The stellar distribution becomes more spherical,called the bulge, closer to the centre. Surrounding the disk is a halo of older stars and globularclusters, the latter containing typically 105 stars each. An even larger distribution of dark matter isbelieved to encompass this visible material.

are not. Moreover, a study of the bending of light from background sources, called

gravitational lensing (see description in Sect. 7.3.1) confirms the presence of dark

matter in clusters (Figure 7.6) and elsewhere.

Many sensitive searches to detect dark matter have been carried out under the

assumption that at least some of the material is baryonic. Baryonic matter is any

matter containing baryons, which are protons or neutrons2. This includes all elements

with their associated electrons as well as any object that is thought to have been

baryonic in the past3. For example, if there are large spherical distributions of dark

matter around galaxies, then such a distribution should also be present around our own

galaxy, the Milky Way (Figure 3.3). Much observational effort has therefore been

expended to measure the gravitational lensing from baryonic objects in the halo of the

Milky Way, be they truly dark or just too dim to be directly detectable. The most likely

candidates are objects that did not have sufficient mass to become stars (like Jupiter), or

the remnants of stellar evolution such as white dwarfs, neutron stars, and black holes

(see Sects. 3.3.2 and 3.3.3 for a description of these objects). These have been

collectively nicknamed massive compact halo objects (MACHOs). The result of

such efforts (see Sect. 7.3.2) is that the masses of all MACHOs fall far short of what

is required to account for the dark matter.

The failure of observations to detect baryonic darkmatter either directly or indirectly

as well as other cosmological constraints (see below) have led to the current view that

most dark matter consists of unusual matter. Since the dark particles have been so

elusive, they must not radiate or interact very easily with other matter, except via

gravity. This rules out most known non-baryonic particles. Also, since dark matter does

gravitate and has been present since the time the galaxies and stars were forming, it

must also have played a part in determining the scale over which structures were

established in the Universe. Non-baryonic particles such as neutrinos, therefore, could

not make up most of the dark matter because they move at relativistic speeds and the

clusters of galaxies that form in a universe dominated by such particles would be too

large in comparison to what is observed. These criteria suggest that most of the dark

matter must consist of particles not yet observed in Earth-based laboratories or

accelerators; such particles are collectively referred to as weakly interacting massive

particles (WIMPs). Some are predicted by well-established theory, others are more

ad hoc. But since classical telescopes have been unable to detect radiation from dark

matter, the new ‘telescopes’ are nowWIMP-seeking particle detectors in underground

labs4.

The search for dark matter is no less than a search for most of the mass of the

Universe. Unless there is something seriously wrong with our understanding of gravity

and current cosmological models, it is estimated that 85 per cent of the matter in the

2There are other types of baryons as well but they are unstable, with lifetimes of no more than about 1010 s.3An example of an object that is no longer baryonic is a black hole whose precursor was a massive star(Sect. 3.3.2).4One example is the Project in Canada to Search for Supersymmetric Objects (PICASSO), which is in the labs ofthe Sudbury Neutrino Observatory (Figure I.3.a), 2 km below ground.


Universe is dark and of unknown nature (see Figure 3.4). Only 15 per cent5, a value set

by knowledge of Big Bang nucleosynthesis, is believed to be baryonic (from Refs.

[171], [109], [16]) and potentially visible via radiation. (Note that the energy density of

radiation itself constitutes a tiny fraction of the energy density of matter and is included

in the matter fractions quoted here.) Of this 15 per cent, however, only half again is

accounted for by direct observation, although there is evidence that the remainder may

be in hot, intergalactic gas that has been difficult to observe in the past (Ref. [112] and

Sect. 8.2.3). Thus, ‘missing matter’ exists on several levels, from some missing

baryonic matter, to the missing non-baryonic matter, the latter presumably in WIMPs.

What is even more astonishing, however, is that recent discoveries (see Appendix F)

imply that the Universe is dominated, not bymass, but by a mysterious dark energy. All

mass, both light emitting matter and dark components, make up only 27 per cent of the

total energy content of the Universe. The wonders that we observe in the Universe,

from g-ray bursts, to massive amounts of hot X-ray emitting gas in galaxy clusters, to

the myriads of galaxies we see optically in the near and far Universe (Figure 3.1), to

cold radio-emitting dust clouds, and all radiation – amount to, at most, a tiny 4 per cent

of this total energy (Figure 3.4). The revelation as to what this dark energy and dark

matter really are, when it finally comes, promises to revolutionize our understanding of

the Universe, and possibly of Physics itself.

Figure 3.4. Estimates of the components of the energy density of the Universe from a variety ofsources (Sect. 3.2). ‘Energy density’ refers to all forms of energy. ‘Dark Energy’ is the component, asyet unknown, that causes the apparent acceleration of the Universe. ‘Matter’ includes all forms ofmatter, both light and dark, as well as all radiation, the latter being a small fraction at the presenttime. The values for dark energy and matter (first line) are thought to be known to within a fewpercentage points. The uncertainty on what fraction of the baryonic matter is considered to havebeen detected is larger. The percentages of the total are referred to as O in Appendix F.

5This value could be 18 per cent. There is some variation in these quantities, depending on the specificobservations.

3.2 DARK AND LIGHT MATTER 69

3.3 Abundances of the elements

3.3.1 Primordial abundance

Of the small fraction of the Universe that makes up all of the stars, galaxies, and other

objects that we see today, we still need to account for its constituents. The abundances

of the elements can be quantified by their number fraction (Figure 3.9) or by theirmass

fraction. The latter is normally expressed as the mass fraction of hydrogen, X, helium,

Y, and all heavier elements, Z, respectively,

X MH

MY MHe

MZ Mm

Mð3:1Þ

where MH, MHe and Mm are the total masses of hydrogen, helium, and all other

elements, respectively, and M is the total mass of the object being considered. Thus,

Xþ Yþ Z ¼ 1 ð3:2Þ

Since the mass fraction of all elements other than hydrogen and helium tends to be

quite low, these are collectively referred to as metals and Z is referred to as the

metallicity.

In the immediate aftermath of the Big Bang, only hydrogen (H), helium (He), and

trace amounts of the light elements, deuterium (D), helium-3 (3He) and lithium-7 (7Li)

were formed6. Although the temperature was extremely high, the expansion of the

Universe was so rapid that the density quickly became too low to drive the nuclear

reactions required to create heavier elements. The abundances at this time are said to be

primordial with values (Ref. [114]) of,

X ¼ 0:77 Y ¼ 0:23 Z ¼ trace ð3:3Þ

3.3.2 Stellar evolution and ISM enrichment

If the Big Bang could not produce the heavier elements, how then were they formed?

The answer lies within stars. As the Universe expanded, inhomogeneities in the matter

density resulted in the collapse and formation of the early protogalaxies and the stars

within them. The process of star formation involves primarily a gravitational collapse of

a molecular gas cloud, but the details of star formation are not entirely understood.

Besides gravity, it is likely that turbulence also plays an important role (Ref. [35]).

6The numbers specify the atomic weight; for example, normal helium has two protons and two neutrons in thenucleus, whereas helium-3 has two protons and only one neutron. Deuterium is an isotope of hydrogen with oneproton and one neutron in its nucleus. An isotope of an element has the same number of protons but differentnumbers of neutrons in the nucleus.


As stars form, the collapse and compression of matter produce an increase in interior

temperature and density. If the mass of the object is less thanMmin¼ 0:08M, then the

interior temperature does not rise sufficiently to ignite nuclear burning (thermonuclear

reactions) in the core. Such objects are referred to as brown dwarfs – would-be stars that

have never ignited. For higher mass objects, however, once the core temperature rises to

a sufficiently high value, (of order 107 K) the first thermonuclear reactions can begin.

These reactions slowly convert the lighter elements into heavier ones through fusion.

The energy released by this process both produces the pressure required to halt any

further collapse and also causes the star to shine. A current estimate of the star

formation rate in the Milky Way is 7.5 stars per year, corresponding to a mass of

4M per year (Ref. [49]).

Low mass stars (masses less than about 1.5 M) convert hydrogen into helium in

their cores by a series of steps called the PP chain (Figure 3.5) which ultimately turns

four protons into a helium nucleus. This is the dominant energy generation mechanism

1H + 1H

2H + 1H

3He + 3He 4He + 2 1H

3He + γ

2H + e+ + ve

+ ve

3He + 4He 7Be + γ

8B + γ

24He8Be

24He

(PP II)

(PP I)

(PP III)

99.7% 0.3%

69% 31%

7Be + H7Be + e– 1

7Li + H1 8Be + e+

7Li + ve

8B

Figure 3.5. The P–P chain of nuclear reactions, illustrated here, powers all stars less than about1.5 M. Note that the intermediate elements created are used up again so the only product is 4He.Two branching arrows indicate that two possible reactions can occur with the probabilities noted.However, the reaction in the PPI branch occurs more quickly than the precursor step to PPII andPPIII so, for the Sun, the production of 4He ends via the PPI branch 86 per cent of the time. Notethat the first two reactions need to occur twice in order to drive the PPI reaction. In the Sun, thePPI chain can be summarized as 4H ! 4He þ 2eþ þ 2ne plus 26 MeV of thermal energy. Of the non-nuclear photons or particles, g represents a g-ray photon, eþ is a positron and ne is an electronneutrino. The neutrino in the PPIII chain has historically been an important player in the so-calledsolar neutrino problem in which fewer electron neutrinos from the Sun were observed thanpredicted by the above steps. This problem was solved in the year 2002 with the definitivedetection of different ‘flavours’ (types) of neutrino by the Sudbury Neutrino Observatory (FigureI.3.a), showing that neutrinos can undergo oscillations and change flavour en route to the Earth(Ref. [106]).

3.3 ABUNDANCES OF THE ELEMENTS 71

in the Sun, for example, where it takes place in a core region of radius, 0:2R. Sincethe mass of a He nucleus is slightly less than the mass of four free protons, (4.0026 uamu

compared to 4.0291 uamu, Table G.2), there is a net mass loss that occurs in the process.

This mass loss has an energy equivalent (DE ¼ Dm c2) and it is this energy that powers

the stars (Example 3.1, Prob. 3.1). The energy, DE, is referred to as the binding energy

of the nucleus since ‘unbinding’ the nucleus would require this amount of energy. The

longest period of a star’s stable lifetime is spent in core hydrogen burning and therefore

the timescale set by this process sets the ‘lifetime’ of a star. It also defines the main

sequence which can be seen as the region from upper left to lower right in the HR

diagram in Figure 1.14. On the main sequence, more massive stars are both hotter and

more luminous, placing them at the upper left of the figure. More massive stars on the

main sequence also burn hydrogen into helium in their cores, but the dominant

reactions are via a series of steps that are different from P–P chain. The result, however,

is the same. On the main sequence, all stars convert hydrogen into helium in their cores.

Data for main sequence stars are given in Table G.7. Main sequences stars are also

called dwarfs because a star on the main sequence is the smallest it will be compared to

later stages in its evolution.

Example 3.1

Estimate the main sequence lifetime of the Sun by assuming that 15 per cent of its mass isavailable for core hydrogen burning. Assume that the luminosity is constant with time.

The available mass isMa ¼ 0:15XM, where X ¼ 0:707 (Eqs. 3.1, 3.4) is themass fraction

of hydrogen in the Sun, so the available number of protons is Na ¼ 0:11M=mp, where mp

is the mass of a proton. The number of reactions is the number of available protons divided

by the number of protons per reaction, Nr ¼ Na=4 ¼ 0:11M=ð4mpÞ, and the total energyreleased is the number of reactions times the energy released per reaction,

E ¼ Nr DE ¼ ½0:11M=ð4mpÞðDm c2Þ. Using Dm ¼ 4:4 1026 g and evaluating for

the Sun (Table G.3), we find E ¼ 1:3 1051 erg. Then the lifetime for constant luminosity

is t ¼ EL

¼ 3:4 1017 s or about 10 Gyr.

The conversion of hydrogen into helium in the core of a star continues until the

hydrogen fuel in the core is exhausted. A thick, hydrogen-burning shell then exists

around an inert He core. The inert central core, having lost a source of pressure, begins

to contract which in turn causes it to heat up. The heating of the core then pumps some

extra energy into the hydrogen-burning shell around it. This additional energy is

sufficient to produce greater pressure in the shell-burning region which then acts

upon the non-burning outer layers of the star, causing them to expand. The resulting

increase in the size of the star makes it more luminous and the expansion produces

cooling of the outer surface. The surface of the star is what is actually observed, so the

star moves upwards and to the right in the HR diagram (Figure 1.14), becoming a red

giant. This is the cause of the main branch from the main sequence towards the upper


right in Figure 1.14. Data for giant stars are given in Table G.8 and a theoretical HR

diagram with the evolutionary track of a solar-mass star is shown in Figure 3.6.

As the star becomes a red giant, the He core is compressing, its temperature rises,

and the core density can become very high. For stars less massive than about 2:3M the

dense He core becomes electron degenerate. This means that the electrons fill the

available free energy states, starting from the lowest possible energies and filling each

higher energy state sequentially. Only one electron can occupy any given quantum state

(the Pauli Exclusion Principle, Appendix C.2) and the occupation of these states

becomes independent of temperature. If the atomic nuclei, and the electrons with

them, are packed very closely together (small Dx) in the dense core, the Heisenberg

Uncertainty Principle (Table I.1) states that the uncertainty in the momentum of the

electrons is very high (large Dp). If the uncertainty in the momentum is high, then the

momentum itself must be high and so must be the electron velocities. With sufficiently

high electron velocities, the electrons themselves can supply enough pressure to hold

up the core, dominating over the particle pressure produced by the random motion of

nuclei (see Sect. 3.4.1). A typical state of such material is one of extreme density, with

the mass of the core containing up to 1.44 M within a volume approximately equal to

the size of the Earth.

For stars with masses between about 0.5 M and 2:3M, the core temperature

eventually reaches a sufficiently high value (108 K) that He burning is ignited. He

burning under degenerate conditions occurs quickly (within hours) and is called theHe

Time spent :

Age :

104

103

102

10

1

.1

.01100,000 20,000

WDcoolingcurve White dwarf

Main sequence

Red giant

Red supergiantPlanetary nebulaform

He coreburning

Sun now

Core star cools

Core star shrinks. Ejected gases

MAIN SEQUENCE

10,000 5,000 3,000

Temperature

Lum

inos

ity (

sola

r un

its)

~9 billion yrs ~1 billion yrs ~100 million yrs ~10,000 yrs

4.5 billion yrs (now)

Main sequence Red giant Yellow giant Planetary nebula White dwarf

12.3306 billion yrs12.3305 billion yrs12.3 billion yrs12.2 billion yrs

4.5 billion years

H core burning

12 billion years

H shell burning

He shellburning

12.2 billion yearsHelium flash

Outer layers ejected

Figure 3.6. Evolutionary track of a 1 M star in the HR diagram (see Figure 1.14 for anobservational analogue). (Adapted from http://skyserver.sdss.org/edr/en/astro/stars/stars.asp)


flash, although the rapid flash occurs within the star and therefore is not directly

observable. The star, however, takes a sharp change of direction on the HR diagram

(see Figure 3.6) as a result. The onset of He burning lifts the degeneracy, the core then

stably burns helium, converting it to carbon7, and the outer layers contract again.

Hydrogen burning will still be occurring in a shell around the He-burning core so stars

start to become differentiated, with carbon in the core, helium farther out, and

hydrogen in the outer layers. A similar evolution results for stars of mass higher

than 2:3M except that helium burning begins before the core becomes degenerate so

there is no internal helium flash. For stars of mass lower than about 0:5M, thetemperature is not expected to become high enough to ignite helium burning. By the

red giant phase, these stars have virtually reached the ends of their lives and should

expel their outer layers as described below. However, the main sequence lifetimes of

such low mass stars are greater than the age of the Universe (Prob. 3.1)! Therefore, we

have not yet seen the end products of stellar evolution for stars of such low mass.

The remaining evolution of stars depends on their mass. Stars like our Sun

(Figure 3.6) eventually exhaust the helium in the core and evolve to the right again

into a red supergiant phase (see Table G.9 for data on supergiant stars). The internal

and external behaviour mimic the first hydrogen core exhaustion stage but now the

carbon core becomes degenerate. The star will then go through a stage of mass loss via

strong stellar winds, pulsating instability, and/or other forms of mass loss (see Sect.

5.4.2). Since the greatly expanded outer layers are tenuous and weakly bound

gravitationally, they are easily expelled, forming a nebula that can take on a rich

variety of morphologies. These are called planetary nebulae8, examples of which are

shown in Figure 3.7. The complex nebular morphologies are not completely under-

stood but seem to be related to different phases of mass loss. Circumstellar disks,

magnetic fields and the presence of companions may also play a role (Ref. [93]). What

remains behind is the hot degenerate core at the centre of the nebula which is now

referred to as a white dwarf.

Stars more massive than the Sun can continue to go through a slow series of

expansions and contractions in response to what is occurring inside them and what

temperatures, and subsequent reactions, are achievable. This results in evolutionary

tracks on the HR diagram that include repeated motions to the right or left in

comparison to what is shown in Figure 3.6. Eventually, all stars with masses less

than 8M, which amounts to 95 per cent of all stars that have evolved off the main

sequence during the lifetime of the Milky Way (Ref. [93], Prob. 3.3), go through the

planetary nebula stage, leaving a white dwarf core behind. The white dwarf may

contain both carbon and oxygen, but it will never be more massive than 1:44M, avalue called the Chandrasekhar limit after the Indian astrophysicist who determined

this maximum value (electron degeneracy pressure is insufficient to support a mass that

7This is called the triple alpha process because three He nuclei, also known as a particles, eventually become acarbon nucleus.8The term ‘planetary’ is used for historical reasons. The first planetary nebulae that were discovered were of smallangular size and showed a blue-greenish hue which was similar to the colour of the outer planets, Uranus andNeptune.


is higher than this). As indicated earlier for the degenerate core of a star, a white dwarf

is very dense, its mass having been compressed into a size which is only about the size

of the Earth.

The material expelled as a planetary nebula will be enriched in heavy elements that

have been formed in the interiors of the progenitor stars. From the enriched expelled

material, new stars will form. Thus, as generations of stars are ‘born’ and ‘die’, the

metallicity of the ISM (and of the Universe) increases with time.

3.3.3 Supernovae and explosive nucleosynthesis

A remaining event that contributes to metallicity and is responsible for elements

heavier than iron (Fe) is a supernova explosion9. Stars of mass 08M do not die

out gradually as do lower mass stars, but rather explode, spewing their outer layers into

the interstellar medium (ISM) at speeds of tens of thousands of km/s. This is called a

Type II supernova10, one of the most energetic events in nature (Figure 3.8). A typical

supernova kinetic energy is 1051 erg. Not only does the internal metal-rich material

enter the ISM, but also the explosion itself is so energetic that it drives many more

Figure 3.7. Examples of planetary nebulae, the end result of the evolution of low mass stars like ourSun. On the left is NGC 6751 and on the right is NGC 6543 (the Cat’s Eye Nebula). The colour schemeis: red¼ [N II] (l 6584), green ¼ [O III] (l 5007 A) and blue¼ V band. The square brackets meanthat the spectral lines are ‘forbidden’ (see Sect. 9.4.2). (Reproductions by permission. For NGC 6751:Arsen R. Hajian, USNO, Bruce Balick, U Washington, Howard Bond, STScI, Nino Panagia, ESA, andYervant Terzian, Cornell; For NGC 6543: B. Balick, J. Wilson, and A. R. Hajian) (See colour plate)

9It has also become apparent that massive stars in a very large red giant phase can spew heavy elements moremassive than Fe into the ISM via stellar winds. The heavy elements may dredge up from the interior.10There are two main types of supernovae which occur in different environments and by different processes. Wedescribe only Type II here. The Type I supernovae are more luminous and are used as distance indicators asdiscussed in Appendix F. They also contribute to the abundances of the heavy metals, including those moremassive than Fe.


reactions, a process called explosive nucleosynthesis. It is in this explosive event that

elements heavier than Fe are formed.

The reason for the extraordinarily different fate of high mass stars lies in the nature

of intranuclear forces. The force that holds the nucleus together is called the strong

force and it acts on small ( 1013 cm) scales between nucleons11, i.e. baryons that are

in an atomic nucleus. For light elements with few nucleons, each ‘feels’ the force of

every other nucleon and so as the number of nucleons grows, so does the binding force

per nucleon. The higher the binding force per nucleon, the more stable is the element

and the greater is the mass deficit when it forms. Eventually, however, the nucleus

becomes so large that Coulomb forces (in this case, the electrostatic repulsion between

protons) which act on larger scales, become important. Then as the nucleon number

grows, the binding energy per nucleon decreases and the nucleus becomes less stable.

The consequence of this behaviour is that light elements release energy when under-

going nuclear fusion, such as in stars, and heavy elements release energy during

nuclear fission, such as currently occurs in nuclear reactors on Earth12. The element

Figure 3.8. The galaxy, M 51, before (left) and after (right) a Type II supernova went off in itsdisk. The supernova, named SN 2005cs, was discovered in June, 2005 by Wolfgang Kloehr using an8 inch telescope, and can be seen directly below the nucleus in the spiral arm closest to thenucleus. A supernova explosion takes only a fraction of a second but maximum light is achieved 2to 3 weeks later. The brightness of a Type II may then plateau or decline in brightness over about80 days followed by a continuing slow decline thereafter. This galaxy is 9.6 Mpc (3:3 106 ly)distant so the supernova detected in 2005 actually occurred 3.3 Myr earlier. (Reproduced bypermission of R. Jay GaBany, Cosmotography.com)

11The strong force actually acts between quarks that are the constituents of nucleons.12The first steps towards creating a fusion reactor have been taken in the establishment of ITER (Latin for‘the way’), an international project located in France. Because only low pressures, in comparison to the centre ofthe Sun, are possible, ITER will operate at 1:5 108 K, an order of magnitude higher than the temperature at thecentre of the Sun! If successful, future reactors will be like ‘mini-Suns’, available for power.


at which the binding energy per nucleon is highest and at which this transition occurs

is iron13.

For stars of mass 08M, the core temperature achieves a value high enough to

initiate fusion reactions of Fe. However, rather than releasing energy during this

reaction (called exothermic), the reactions now require energy (called endothermic).

With no energy source in the core and available energy going into the endothermic

reaction, the core pressure is removed and the core instantly implodes. The result of the

core implosion is a remnant, either a neutron star, a star so dense that the mass of a Sun

or more has been compressed into a volume only tens of km across14 or a black hole, a

region of space around a singularity which contains the mass of several Suns15.

Although the details of supernovae are not perfectly understood, it is believed that

the explosion of the outer layers is enabled by shock waves16 that are triggered by the

core collapse. The resulting material, enriched in heavy elements through explosive

nucleosynthesis, contributes to the metallicity of the interstellar medium (ISM).

The supernova itself can become extremely bright (Figure 3.8), sometimes outshining

the light from the entire parent galaxy. Its light then declines in a characteristic

exponential fashion with time, though changes in slope can also occur (see also

Sect. 9.1.3). Such a plot of a variation in light with time is called a light curve.

Eventually, the exploded outer layers become visible as a supernova remnant, such as

the one shown in Figure 1.2.

3.3.4 Abundances in the Milky Way, its star formation history and the IMF

The elemental abundances of our Solar System, expressed as a number fraction,

measured from Solar spectral lines (see Figure 6.4) and meteorites, are shown in

Figure 3.9. This plot represents the abundances in the local Galactic neighbourhood at

the time the Solar System formed. The values thus reflect the origin of the light

elements in the Big Bang as well as subsequent evolution or other processes that could

alter these relative abundances since that time. Enrichment of heavy elements by

previous generations of stars is clearly the most important effect, the current supernova

rate for the Milky Way being estimated at about two per century. However, other

processes are also involved. For example, high energy cosmic rays (Sect. 3.6) can

collide with nuclei, breaking up heavier nuclei into smaller pieces, a process called

spallation. In addition, the stability of nuclei against radioactive decay will also affect

13The most tightly bound element is actually 62Ni, but this element is not very abundant and therefore does notplay an important role in supernova explosions.14In such a remnant, the density is so great that electrons have combined with protons to produce a ‘sea’ ofneutrons; the density of a neutron star is approximately equal to the density of a single neutron and the star is heldup from further collapse by neutron degeneracy, the nuclear version of electron degeneracy. There is someuncertainty about the upper limit to the mass of a neutron star but a reasonable estimate is 3M.15For larger core masses, there is no known force that can withstand a collapse to this state.16Shock waves are sharp, rapidly moving density discontinuities that are produced when the motions of particlesare faster than the speed of sound in the medium.


these abundances. Consistent with an origin in the Big Bang and in spite of all

subsequent evolution, the abundance of H is still, by far, greater than all the other

elements combined, constituting approximately 90 per cent, by number, of the Uni-

verse. Normalized to a hydrogen abundance of 1, He constitutes only 8.5 per cent, and

Li is a mere 109 per cent17. Even the stable Fe (atomic number¼ 26) has an

abundance only 3 103 per cent of hydrogen. The most abundant of the metals is

oxygen at 6:8 102 per cent. Given the dominance of hydrogen in our Universe, it is

important to understand this simple atom extremely well. A review of the properties of

hydrogen is given in Appendix. C.

The corresponding mass fractions for the abundances illustrated in Figure 3.9

(Ref. [44]), referred to as Solar abundance, are,

X ¼ 0:707 2:5% Y ¼ 0:274 6% Z ¼ 0:0189 8:5% ð3:4Þ

From measurements of metallicities in interstellar clouds and other stars, Solar

abundances appear to be typical of abundances elsewhere in the disk of our Milky

Way.

As has been seen with the production of heavy elements within stars, there

are many processes involved in the slow conversion of the primordial abundances of

17Li tends to be destroyed in stellar nuclear reactions.

0

2

4

6

8

10

12

14

741 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91

Atomic Number

log

(abu

ndan

ce) +

12

Figure 3.9. Logarithmic plot of elemental abundances in the Solar System as a function of atomicnumber. Black boxes represent values for the Sun and clear box values are for meteorites. In severalplaces there are no data, for example a measurement for H in meteorites or for short-livedradioactive elements. All abundances have been normalized to hydrogen which has a value of 12 inthis plot. For example, a value of 6 on the graph corresponds to an abundance of 106 with respectto hydrogen. Abundances refer to the numbers of particles, rather than their weights. Data fromRef. [154].


Eq. (3.3) to the Solar abundances of the Galactic disk. The increase in metallicity

depends on the amount, rate, and metallicity of material expelled into the ISM by

planetary nebulae and supernovae and the heaviest elements originate in supernovae

and the winds of massive red giants. Since the end points of stellar evolution depend on

stellar mass, it is important, then, to ask how many stars in different mass ranges are

actually formed.

The admixture of stars of different masses, when first formed, is called the initial

mass function (IMF). Various IMFs have been proposed in the astronomical literature,

attesting to the difficulty in obtaining this function and its possible variation with

environment. Good examples are shown in Figure 3.10. The functional forms for the

–7

–6

–5

–4

–3

–2

–1

–1 –0.5 0 0.5 1 1.5 2

x = log(M)

log

(dn

/dx)

Disk

Halo

IMF

Figure 3.10. The initial mass function (IMF) for the disk and stellar halo of the Milky Way. Massesare in units of M. The IMF, given by dn=d[log(M)] (pc3 ½logðMÞ1) represents the numberdensity of stars per logarithmic mass interval that have formed over the history of the region beingstudied. It is determined by counting the number density of stars in a given luminosity interval,applying corrections for observational biases, converting to the number density of stars in a givenmass range, and then applying corrections for the number of stars that have evolved, i.e. that areno longer on the main sequence. The functional expressions are provided in Eqs. (3.5) and (3.6).There are some significant error bars (not shown) associated with the fitted parameters, but theplot provides a good idea as to how many stars of different masses form in the Milky Way. There areclearly many more stars per unit volume in the disk than in the halo and many more low mass starsin comparison to high mass stars. Star formation via gravity alone cannot reproduce the details ofthese plots and it is likely that turbulence plays an important role. From Ref. [35].


two curves, applicable to the Milky Way’s disk and halo, respectively, are,

dn

dx

disk

¼ 0:158 exp

1

2

xþ 1:10

0:69

2M < 1:0M

¼ 4:4 102 M1:3 M 1:0M ð3:5Þdn

dx

halo

¼ ð3:6 104Þ exp 1

2

xþ 0:66

0:33

2M < 0:7M

¼ ð7:1 105ÞM1:3 M 0:7M ð3:6Þ

where x ¼ logðMÞ and M is the stellar mass in Solar mass units. The term, dn=dx, inunits of pc3 ½logðMÞ1

, gives the number density of stars per cubic parsec per

logarithmic mass interval (Ref. [35]).

Figure 3.10 illustrates that far fewer high mass stars are formed in comparison to

those of low mass (Example 3.2, Prob. 3.2). However, high mass stars that end their

lives as supernovae also live shorter lives since they burn their energy at a faster rate

while undergoing normal hydrogen fusion (Prob. 3.1). As long as the region of the ISM

is rich in gas and able to form stars, this means that there will be many successive

generations of massive stars forming and dying while low mass stars like our Sun

continue to ‘simmer’ in the stable state of burning hydrogen to helium in their cores.

Therefore, even though there are fewer high mass stars, the massive stars will have the

greatest influence on the metallicity of the ISM. Since high mass stars are short-lived

(of order 106 yr), they will also not stray far from their places of origin during their

lifetimes. Therefore, the presence of high mass stars and any observational conse-

quences of high mass stars alone, such as HII regions (which only exist around hot

massive stars, see Sect. 5.2.2) or supernovae, are taken to be proxies for active star

formation. We say that hot high mass stars, HII regions and supernovae are all tracers

of massive star formation. It is important to note that, once some time has passed, the

admixture of stars in a region will not be the mixture of Figure 3.10 because of the

different rate of evolution of stars of different masses. Rather, the admixture of stars is

determined by noting which stellar mix results in the observed HR diagram for a given

region.

Example 3.2

How many disk stars per pc3 with masses between 0.1 and 1 M are initially formed incomparison to those between 10 and 100 M?

From Eq. 3.5 (see also Figure 3.10) with x ¼ log M, in the low mass range,

n ¼ 0:158

Z x¼0

x¼1

exp 1

2

ðxþ 1:10Þ0:69

2 !dx ¼ 0:106 pc3:


In the high mass range,

n ¼ 4:4 102

Z x¼2

x¼1

M1:3 dx ¼ 4:4 102

Z x¼2

x¼1

101:3x dx ¼ 7:00 104 pc3

Therefore, there are 151 times more stars formed per cubic parsec in the low mass range

than the high mass range.

Abundances in ourMilkyWay are very important probes of its star formation history

and, by implication, its global formation and evolution. For example the IMF

(Figure 3.10), the quantity of gas present, and knowledge of stellar evolution allow

models to be built that predict the metallicity distribution in a given environment now.

The modelled result can then be compared to that observed. If one assumes an initially

lowmetallicity and that the amount of gas in a given region of the Galaxy remains at its

location without flowing out or other gas flowing in (called the closed box model), then

one finds that too few stars of low metallicity are observed in comparison to what the

models predict18. This means that somemodification is necessary in the assumptions of

the model. It would appear that some additional enrichment of metals is required,

possibly from inflowing gas, or possibly frommetal-rich gas from previous generations

of stars that had stronger stellar winds and/or mass loss than expected.

When we look at the metallicity of stars in the disk in comparison with the halo,

another important result emerges. Away from the gaseous, dusty spiral-like disk within

which most of the visible material of our Galaxy is found, there is a more spherical halo

of stars surrounding the disk in which there is very little gas and no cold dense gas from

which stars can form (see Figure 3.3). The metallicities of halo stars vary, but a typical

value is Z ¼ 0:0005, less than 3 per cent of the disk value. Clearly, the metallicity in the

halo is very low in comparison to the disk and this provides a clue as to the comparative

star formation history of these two regions.

The much lower density of stars, as shown by the IMF (Figure 3.10) also attests to

the very different environment and history of the halo. Star formation in our Galaxy

has mostly occurred and is still occurring in its disk but it is no longer occurring in its

halo which now contains the oldest stars in the Galaxy. Although the metallicity in

the halo is very low, it is not zero, indicating that some star formation has indeed

occurred there in the past. In fact, similar results are found for other galaxies and

imply that a period of rapid star formation and enrichment occurred very early in the

galaxy formation process when smaller systems were joining to form larger ones

(Figure 3.1). The thin galaxy disks that we see today with their beautiful spiral

structure (Figures 2.11, 3.8) likely formed later when baryons settled into the central

regions. Star formation then continued in the disk, forming a denser stellar environ-

ment there (Prob 3.2). A more complete review of galaxy formation can be found in

18This is called the G dwarf problem.


Ref. [59] and a simulation of the early structures that evolve into galaxies can be seen

in Figure 5.7.

Dynamics, global structure and stellar distribution all provide information about the

formation of galaxies, but abundance is a key component. There is much that is still not

fully understood about the formation process, and the fact that dark matter, of which we

know little, is the dominant component is a strongly complicating factor. Nevertheless,

even the tiniest of visible constituents and apparently insignificant observations such as

metallicity variations can offer just the insights required to piece together a coherent

picture of Galactic history.

3.4 The gaseous universe

The two most abundant elements in our Universe, hydrogen and helium, are gases.

Only under extreme pressure, such as at the centres of planets like Jupiter, can

hydrogen exist as a metallic-like liquid. It follows, then, that most of the Universe is

in a gaseous state. Indeed, most of the metals are also gaseous, given the pressures and

temperatures under which they may be found, for example, silicon within the Sun,

gaseous carbon in interstellar clouds, or iron atoms in hot tenuous gas. Only a tiny

fraction of the observed Universe is in the solid state in the form of dust particles and

the planets, asteroids, and comet nuclei that we see in our own Solar System or infer to

be in others. Once established, the internal tensile strength of solids can keep them in

this phase even in the low pressure environment of space. Ices of various molecules

such as water, carbon dioxide, and methane, are also a common constituent of solid

Solar System material, especially in comets or on cold bodies, distant from the Sun.

The liquid phase, however, appears to be rarest. The Earth has a special place in our

Solar System in that its temperature and surface pressure allow water to exist in all

three phases on its surface. There is nowmounting evidence that liquids can flow on the

surfaces of other planets and moons, for example, via volcanic flows. Liquid water may

leak or be vented from under the icy surfaces of Europa, a moon of Jupiter, and

Enceladus, a moon of Saturn. And liquid water may have flowed on Mars. However,

conditions are not now suitable for long–lived ‘lakes’ or ‘rivers’ on these objects.

A direct observation of stable surface liquids is still, therefore, elusive. The only other

known body on which surface liquid may be present is Titan, another moon of Saturn

and this liquid would be methane (see Figure 3.11).

The gases of which the Universe is largely composed can show a wide variety of

properties, especially of temperature and density. A cold, dusty dense molecular cloud,

for example, is quite different from the hot upper atmosphere or the deep interior of a

star. Table 3.1 gives some typical densities and temperatures for various astrophysical

gases, and images of a few different kinds of gaseous objects that can be found in the

ISM are shown in Figures 1.2, 3.7, and 3.17. Aside from density and temperature,

another important difference is the state of ionization. Gases can be neutral or ionized

or have some partial ionization. A neutral gas is one in which the atoms have their

electrons still attached or bound to them so that there is no net charge on the atom. In


Figure 3.11. On January 14, 2005, the European Space Agency’s Huygens space probe took thesepictures of the surface of Titan while descending to its surface. Titan is Saturn’s largest moon andthe one shrouded in a thick, opaque atmosphere that is predominantly nitrogen with 5 per centmethane (CH4). Since the surface temperature and pressure on Titan is T ¼ 94 K and P ¼ 1:6 atm,respectively, and since the triple point (the temperature and pressure at which all three phases canexist) for methane is T ¼ 91 K, P ¼ 0:12 atm, conditions on Titan appear favourable for methane toexist in the solid, liquid and gas phases. The left image was made from a number of images takenfrom a height of about 8 km and have a spatial resolution of about 20 m. The ‘shoreline’ shown inthis image separates two regions, both of which are believed to be solid. The right image showstantalizing evidence that liquids have at some time flowed on the surface, possibly recently andpossibly seasonally. Newer images of the north polar region of Titan also suggest that ‘lakes ofmethane’ may be present (Ref. [26]). (Reproduced by permission of ESA/NASA/Univ. of Arizona,Brown, D., Hupp, E., and Martinez, C., 2006, NASA News Release: 2006/097)

Table 3.1. Sample densities and temperatures in astrophysical gasesa

neb nm þ nH

c T

Location (cm3) (cm3) (K)

Interplanetary Space 1 104 0 102 103

Solar corona 104 108 0 103 106

Stellar atmosphere 1012 0 104

Stellar interiors 1027 0 107:5

Planetary nebulae 103 105 0 103 104

HII regions 102 103 0 103 104

Interstellar spaced 103 10 (avg: 0:03) 102 105 (avg: 1) 102

Intergalactic space < 105 0 105 106

Intergalactic HI cloudse 106 103 103 105

aRef. [96] unless otherwise indicated.bElectron density. For a pure hydrogen gas that is completely ionized, ne ¼ np where np is the proton density.cnm is the molecular gas density (predominantly H2) and nH is the neutral atomic hydrogen (HI) gas density.dThe temperature quoted here is typical of interstellar HI clouds. However, there is a wide variety of densities,temperatures and degrees of ionization in the interstellar medium. For example, molecular clouds tend to havelow temperatures (5–200 K, typically 30 K) and high densities (102–105 cm3, typically 104 cm3). On the otherhand, for ionized diffuse intercloud gas at ne 0:03 cm3 the temperatures are much higher ( 104 K).eTypical density range for the Ly a forest (see Figure 5.6 and Ref. [129]) for a redshift (Sect. 7.2.2) of z ¼ 3:1.The ionized fraction is not easily determined. Values will vary with redshift.

3.4 THE GASEOUS UNIVERSE 83

gases with higher states of ionization (partially ionized) some atoms will have lost

electrons, becoming ions with a net charge. A highly ionized gas is referred to as a

plasma in astrophysics. Since hydrogen is the most abundant atom, a plasma will have

approximately equal numbers of free protons and free electrons, with some variation

depending on the admixture of other elements and their state of ionization. The degree

of ionization of any given element is specified by Roman numerals after the element

name, where the number of electrons removed is given by the Roman numeral minus

one. For example, neutral hydrogen is referred to as HI and ionized hydrogen is HII.

For metals, there are many more electrons that could leave the atom. Thus, one can

have, for example, neutral carbon, (CI), singly-ionized carbon (CII), or triply-ionized

carbon (CIV). Many such ions have been observed in various astronomical objects,

including some with extremely high states of ionization. For example, iron with all

electrons removed but one (Fe XXVI) has been observed in the spectra of some highly

energetic X-ray sources (e.g. Ref. [40]), including the Solar corona19 (see Figure 6.7).

Important goals in astrophysics are to determine the densities, temperatures, and

ionization states of gases which include, of course, the atmospheres and interiors of

stars. With sufficient information, the energetics involved in maintaining these gases in

the observed states can then be determined. Armed with this knowledge, it may be

possible to piece together a picture of the formation and evolution of the objects. Thus,

there is considerable importance in understanding the physics of gases, especially of

hydrogen. Appendix C outlines the basic physics of the hydrogen atom and the sections

that follow highlight some relevant astrophysics of gases.

3.4.1 Kinetic temperature and the Maxwell–Boltzmann velocitydistribution

Consider a gas in which energy has been exchanged between particles via elastic

collisions. This means that only kinetic energy is exchanged; there are no radiative

energy losses, ionizations, or other processes occurring between the particles. In such a

case, a velocity distribution is set up such that the number density of particles in each

velocity interval is described by the Maxwell–Boltzmann, or Maxwellian velocity

distribution. The functional form of this distribution was presented in Eq. (0.A.2) and

an example is plotted in Figure 3.12.

It is theMaxwellian velocity distribution that defines the kinetic temperature, written

as Twithout a subscript, of a gas. For particles of mass, m, T is related to the average of

the squares of the particle speeds, or the mean-square particle speed, hv2i,

1

2m hv2i ¼ 3

2k T ð3:7Þ

19The Solar corona is the faint, low density, outermost region of the Sun’s atmosphere, visible to the eye during atotal Solar eclipse. Coronal temperatures are typically 106 K but can be higher during times of high activity(Sect. 6.4.2).


The root-mean-square particle speed is then vrms ¼ffiffiffiffiffiffiffiffiffihv2i

p. It can be shown (Prob. 3.4)

that the most probable speed is slightly lower,

vmp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 k T=m

pð3:8Þ

and the mean speed is higher,

hvi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8 k T=m

pð3:9Þ

The left hand side of Eq (3.7) represents the mean kinetic energy of the particles, hEki,so T is a measure of the mean kinetic energy of particles in a gas. A gas that can be

0

1e–06

2e–06

3e–06

4e–06

5e–06

6e–06

100000 200000 300000 400000 500000

Maxwell-Boltzmann Velocity Dist’nn = 1 cm^–3, T = 100 K, m = m_H

Velocity (cm/s)

n(v

) (c

m^-

3/(c

m/s

))

Figure 3.12. Maxwellian velocity distribution for neutral atomic hydrogen at T ¼ 100 K and atotal density of 1 cm3. The ordinate specifies the number of particles in each velocity interval andthe abscissa gives the velocity. The area under this curve returns the total density. See Eq. (0.A.2)for its functional form.


described by a single kinetic temperature is said to be in thermal equilibrium20. Such a

result would occur if a hot gas and a cold gas were set beside each other in an insulated

box. Eventually both gases would reach a single warm temperature.

The Maxwell–Boltzmann distribution can be derived from statistical mechanics

which applies statistics to the motions of large numbers of particles in a gas in order to

predict its macroscopic properties. The resulting equation (Eq. 0.A.2) can be seen to

depend on a number of parameters, including the Boltzmann factor,

ex=ðk TÞ ð3:10Þ

where x is an excitation energy. In this case, x ¼ 12m v2 represents the kinetic energy of

a particle. The Boltzmann factor indicates that there is a lower probability of finding a

particle in a given excitation state as the excitation energy increases in comparison to

the mean kinetic energy. In addition, however, the Maxwellian velocity distribution

also depends on the quantity, 4p v2 which represents the surface area of a shell about

an origin in three-dimensional ‘velocity-space’. As the velocity (or kinetic energy)

of a particle increases, the velocity shells become larger, allowing more possible

orientations of the velocity vector of a particle. This means that more ‘states’ are

available to be occupied as the velocity increases. These competing effects result in the

final Maxwellian distribution that first increases as v2, reaches a peak then decreases

exponentially (Figure 3.12). The constant multiplier in the equation is a normalization

factor to ensure that the total number density of particles is correct.

If a gas can be described by a Maxwellian velocity distribution, then there must have

been a sufficient number of elastic interactions that this thermal equilibrium could be

established. How valid is this assumption for typical astrophysical conditions? For

gases in which T is less than about 8 104 K, particle encounters are almost always

elastic and there will therefore be many elastic collisions before an inelastic encounter

occurs (Ref. [160]). The timescale required for establishing an equilibrium tempera-

ture, ttherm, can be determined by asking how long it would take for a faster test particle

(the hotter gas particle) to slow down as it diffuses through the gas of slower particles

(the colder gas particles)21. It can be shown that this timescale is quite short under

astrophysical conditions and depends on the density of the dominant constituent of the

gas. For example, in cool (T ¼ 100K) neutral atomic (HI) clouds in the interstellar

medium, the timescale for establishing an equilibrium temperature via H–H collisions

is ttherm ¼ 16=nH yr, where nH is the hydrogen density. If nH ¼ 1 cm3 (Table 3.1) then

only 16 years are required, a very short time by astronomical standards. For a fully

ionized hydrogen gas at T ¼ 104 K and the same density, the equilibrium timescale for

electron–proton ‘collisions’22 is only ttherm ¼ 8:2 103 s, that is, only a few hours. The

20The term, ‘thermal equilibrium’, is sometimes used differently from here, for example, to represent energybalance within stars.21The equilibrium timescale is one half of the slowing down time since the slower moving particles will alsospeed up. Such a calculation involves, as a starting point, determining the mean time between collisions, anexample of which is provided in Sect. 3.4.3.22For coulomb interactions, a collision is considered to be an interaction that significantly affects the trajectory ofa particle.


timescale for electron–electron collisions is equivalent to that for electron–proton

collisions, provided the electron and proton densities are equal in the gas. Clearly,

higher densities facilitate the process of temperature equalization and Coulomb inter-

actions shorten the timescale even more. In even the low density ISM, then, a

Maxwellian velocity distribution is expected and therefore it should also present in

regions of higher density, such as stellar atmospheres (e.g. densities of order

1012 cm3).

We therefore assume, in most cases, that astrophysical gases are in thermal

equilibrium. Departures can occur, however, especially in gases of very low density.

There may also be other physical effects that inhibit the process of temperature

equalization. If the gases are physically prevented from diffusing together, thermal

equilibrium will not be achieved (for example, magnetic fields can constrain the

positions of charged particles in an ionized gas). Particles whose energetics are driven

by processes that are intrinsically non-thermal in nature, such as cosmic rays (Sect.

3.6), will not achieve a Maxwellian distribution at all. Collisions of gas particles with

dust particles require revisions from the above description as well. Figure 3.13 shows a

situation common for an HII region in which dust is mixed with the ionized gas, yet the

dust temperature may be two orders of magnitude lower than the kinetic temperature

of the gas. Dust temperature, as quoted, is conceptually different from the gas

temperature, however, since it does not refer to the random speeds of the dust particles

but rather a temperature at which the dust grain radiates; this will be further elucidated

in Sect. 4.2.

Figure 3.13. This image shows most of the Lagoon Nebula (also called Messier 8), which is anionized hydrogen region (HII region) visible to the naked eye in the constellation of Sagittarius.The nebula is 1.6 kpc away with an average diameter of 14 pc (there is some uncertainty in thedistance) and the complete nebula appears as large as the full Moon in angular size. The imageshows a complexity of features with colours representing emission from different atoms (oxygen,silicon and hydrogen) and dark bands and filaments representing obscuration by dust. The dust andgas are in close proximity yet the gas temperature is 7500 K whereas dust temperatures are typicallymuch lower, of order 100 K or less. The mean density of electrons in the nebula is ne 80 cm3.(Reproduced by permission of Michael Sherick, Cabirllo Mesa Observatory, 2005) (See colour plate)


3.4.2 The ideal gas

An ideal gas is a gas that obeys the ideal gas law (or perfect gas law, see also

Table 0.A.2 and Prob. 0.3),

Pp ¼ n k T ¼ r

mmH

k T ð3:11Þ

where Pp is the particle pressure, n is the number density (cm3) of particles of any

mass, k is Boltzmann’s constant, T is the temperature, r is the mass density of the gas

(g cm3), and mH is the mass of the hydrogen atom. The quantity, m, is called themeanmolecular weight, defined by

m hmimH

¼ 1

mH N

Xi

mi ð3:12Þ

where mi is the mass of the ith particle, and N is the total number of particles. The

quantity, m, gives the mean mass of free particles in units of mH (Example 3.3). The

mean molecular weight can also be expressed in terms of the abundances of the object.

For example, the mean molecular weights of a completely neutral and a completely

ionized gas are (Prob. 3.5), respectively,

m ¼ 1

Xþ Y4þ Z

Am

ðneutralÞ

m ¼ 1

2Xþ 3Y4þ Z

2

ðionizedÞð3:13Þ

where Am is the mean atomic weight of the metals in the gas. The atomic weight of

particle i, Ai, is related to its mass, mi, by mi ¼ Ai mH. The second expression assumes

that Am=2 1. Therefore, for a completely ionized gas with Solar abundance (Eq.

3.4), m ¼ 0:61. For the particle pressure of electrons alone, Pe, in a fully ionized gas,

the number density of electrons, ne ¼ r=ðme mHÞ, can be used in Eq (3.11) where,

me ¼1

Xþ Y2þ Z

2

ð3:14Þ

Example 3.3

Show that the mean molecular weight of a completely ionized pure hydrogen gas is 1=2.

In such a gas, half of the particles are electrons of mass, me and half are protons of mass,

mp. From Eq (3.12), m ¼ 1mH N

½N2me þ N

2mp 1

mH NN2mp 1

2.


The ideal gas law is derived under the assumption that, when energy is exchanged

between particles, the interactions are elastic. Thus, if the particles were placed in a

small box, the pressure against any side of the box would result from the collective

motions of the particles against the wall, with no corrections required for particle–

particle interactions. The more closely spaced the particles are, the more likely it will

be that non-elastic interactions may occur. Therefore, a common, though not foolproof,

way to decide whether such interactions are negligible is to determine the mean

separation between particles, r, and compare this to the particle radius23, Rp. The

mean separation between particles can be found from nV ¼ N, where n is the gas

density, V its volume, and N the total number of particles, and setting the total number

of particles to 1,

r ¼ 1

n1=3ð3:15Þ

If r Rp, then the gas may be considered ideal. The caution to this rule of thumb is

that other effects may be important, for example, Coulomb interactions between

charged particles, or the fact that the effective volume within which a particle can

move may be reduced due to finite particle size. Another example is the degenerate

cores of low mass stars (Sect. 3.3.2) in which densities are so high that quantum

mechanical effects become important.

The perfect gas law is an example of an equation of state, that is, an equation that

describes how the physical properties of the gas (e.g. pressure, density and tempera-

ture) are related at any location. It is important to note that a complete description of

pressure includes all contributors, including particle pressure (due to the thermal

motion of particles and also free thermal electrons, if present) as well as radiation

pressure (Sect. 1.5),

P ¼ Pp þ Prad ð3:16Þ

If electron degeneracy (Sect. 3.3.2) is an important contributor to the pressure, then it

must be added as well and, if these other terms are significant in comparison to Pp, then

the gas is non-ideal.

3.4.3 The mean free path and collision rate

If a small test particle enters a gas containing larger field particles (the latter initially

considered at rest), there will be some probability that it will collide with a field

particle. The average distance that the particle travels before such an encounter occurs

is called the mean free path,l. The mean free path can be determined by considering a

small volume, V , of the gas, as shown in in Figure 3.14. The probability that the test

particlewill encounter a field particlewill depend on the projected area presented by all

23Comparing to the particle diameter is more accurate, though factors of two are not that important for mostcomparisons, as Example 3.4 illustrates.


field particles,Ap, divided by the total area presented by the region of space, A. If we are

to ensure that an interaction does occur, then the probability is 1 so,

1 ¼ Ap

A¼ N s

pR2¼ nV s

pR2¼ npR2ls

pR2¼ n sl ð3:17Þ

Here N is the total number of field particles, s is the cross-sectional area of each field

particle, and the volume is V ¼ pR2l as shown in the figure. Thus, the mean free path

is given by,

l ¼ 1

n scm ð3:18Þ

If the test particle has a velocity, v, this distance can be expressed asl ¼ vt, wheret ismean time between collisions, thus,

t ¼ 1

n v ss ð3:19Þ

The collision rate for this test particle is just the inverse of the mean time between

collisions,

R ¼ n v s s1 ð3:20Þ

For a more general case with many test particles and field particles, v is the relative

velocity between the test and field particles. For identical particles in a one-temperature

gas, the relevant velocity is the mean particle velocity, hvi, as given in Eq (3.9). If the

Field particles

Test

particle

R

Face-on view Side view

Figure 3.14. A test particle entering a gas containing field particles travels a mean free path(depth of the volume, left) before encountering a field particle. The probability of an encounterdepends on the area presented by the field particles over the total area (right). If the entire cross-sectional area of the cylinder is occupied by particles, then the test particle will certainly encountera field particle.


test particle is of very low mass in comparison to the field particles, such as collisions

between free electrons and protons in an ionized gas24, then v is just the mean particle

speed of the electrons since the protons can be thought of as ‘at rest’ in comparison.

Also, the collision rate,R (Eq. 3.20) refers to the rate at which a single test particle will

encounter another field particle in a gas in which the field particle density is n. If we

wish to calculate the total collision rate in a gas due to collisions by all test particles,

then we need to multiplyR by the total number of test particles, Nt ¼ nt V , where nt is

the test particle density. The total collision rate per unit volume, then, is,

ntot ¼ n nt v s s1 cm3 ð3:21Þ

It doesn’t matter which particle is considered the test particle and which is considered

the field particle as long as the correct values of v and s are used.

If all particles are neutral and can be approximated as elastically interacting spheres,

then s is just the geometrical cross-sectional area of the particles. However, the cross-

sectional area may have to be modified in some circumstances, in which case it is called

the effective cross-section, or seff. For example, seff will differ from the geometric

cross-section if one or both sets of colliding particles are charged (such as electrons

with protons or electrons with electrons) and s may itself be a function of velocity,

sðvÞ. Thus, Eqs. (3.18), (3.19), (3.20), and (3.21) require values of seff that depend onthe type of colliding particles and velocities under consideration. Often, the quantity,

gcoll, where coll refers to the type of collision being considered, is used instead. The

quantity, gcoll is the mean collision rate coefficient which folds together the quantities,

v and s with appropriate weighting. For example, for collisions between neutral

hydrogen atoms, gH-H ¼ hviseff cm3 s1. Some sample values of gcoll and seff for

collisions between various types of particles are given in Table 3.2.

The above expressions are put to use in many astronomical situations. Example 3.4

provides a sample calculation for interstellar HI. Later (Example 5.4) we will provide

an example for electron – proton collisions that result in recombination to HI. A final

important point is that the ‘test particle’ could be a photon, a point we return to in

Example 3.5 and Sect. 5.

Example 3.4

For an interstellar cloud of pure HI at T ¼ 100 K and n ¼ 1 cm3, (a) calculate the geometriccross-section, sg, for H–H collisions. What is the ratio of seff=sg? (b) Determine the mean freepath, mean time between collisions, and collision rate for an H atom. Also, what is the meanseparation between particles. Is this an ideal gas?

(a) Using the Bohr radius, a0 ¼ 5:29 109 cm (Table G.2), sg ¼ p a20 ¼ 8:801017 cm2. By Eq. (3.9), and setting m to the mass of the hydrogen atom,

24See footnote 22 in this chapter.


Table 3.2. Sample collision parameters

Temperature gHH

(K) HI – HIa (cm3 s1)

30 5:1 1010

100 7:4 1010

300 10:2 1010

1000 13:6 1010

HI – HI de-excitationb g21 cm line

(cm3 s1)

30 3:0 1011

100 9:5 1011

300 16 1011

1000 25 1011

electron – proton with recomb.c ar

(cm3 s1)

5000 4:54 1013

10 000 2:59 1013

20 000 2:52 1013

electron – proton without recomb.d seffðcm2)

104 1:4 1015

105 1:4 1017

106 1:4 1019

electron – HI de-excitatione gLya ð2p!1sÞ(cm3 s1)

5000 6:0 109

10 000 6:8 109

20 000 8:4 109

aCollisions between neutral H atoms. Here gHH ¼ hviseff (Ref. [160]).bCollisions between neutral H atoms that result in the de-excitation of thehyperfine ground state (see Appendix C.4).cCollisions between an electron and proton that result in recombination to HI. Inthis case, gcoll is referred to as ar, which is the sum of an;L over all quantum levels,n and L (Appendix C). Case B recombination (see Footnote 6 in Chapter 5 or Sect.9.4.2) is assumed.dCollision without recombination. A collision is taken to be an interaction thatsignificantly affects the trajectory of the electron. Taking e2=reff ¼ me v

2 to be thecondition for a significant deflection (Ref. [160]), we find, seff ¼ p r2eff ¼p ½e2=ðme v

2Þ2. Using Eq. (3.9), this leads to seff ¼ 1:4 107=T2. A moreaccurate derivation would include the cumulative effects of other particles in thegas and would also use the Maxwellian distribution instead of Eq. (3.9). Therefore,these results are meant to be ‘indicative’ only.eCollisions that result in the de-excitation of the Lya line in the permittedtransition (see Appendix C.2) from 2p to 1s only. This has been computed fromEq. (4–11) in Ref. [160] using data from www.astronomy.ohio-state.edu/~pradham/atomic.html. Note that the value for excitation will not be the same asfor de-excitation nor will it be the same for the forbidden transition from 2s to 1s.


hvi ¼ 2:57 105 cm s1. Then hvisg ¼ 2:26 1011 cm3 s1. From Table 3.2, gH-H ¼hviseff ¼ 7:4 1010 cm3 s1, so seff=sg ¼ ðhviseffÞ=ðhvisgÞ ¼ 32:7.(b) From part (a), seff ¼ 32:7 sg ¼ 2:88 1015 cm2. Using this value (or ‘gH-H’ as

required) in Eqs. (3.18), (3.19), (3.20), and (3.15), respectively, we find,l ¼ 3:5 1014 cm

(23 AU), t ¼ 1:4 109 s (43 yr), R ¼ 7:4 1010 s1, and r ¼ 1 cm. The ‘effective

radius’ of the particle is Rpeff¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiseff=p

p¼ 3:0 108 cm which is r so this is an

ideal gas.

3.4.4 Statistical equilibrium, thermodynamic equilibrium, and LTE

Even under conditions in which most interactions between particles in a gas are

elastic and a Maxwellian velocity distribution exists, there will be a minority

of interactions in which kinetic energy is not perfectly conserved and some energy

is exchanged in other ways. A good example is when collisions cause the excitation

or de-excitation of atoms, a process that involves the transition of electrons from

one quantum state to another (see Appendix C). For an excitation, a collision

must exchange an energy that corresponds to the energy difference between

levels. The probability of such a collisional energy exchange will be different

for each transition. Thus, in a Maxwellian velocity distribution, a single value

for v seff, as used in Eqs. 3.19 and 3.20, cannot be used to describe the proba-

bility that a specific exciting or de-exciting interaction will occur. For transitions

between states, i and j, v si;j < v seff since only a fraction of all collisions will

result in a specific transition. Moreover, excitation can also result from the ab-

sorption of a photon and de-excitation can occur spontaneously or via photon

interaction.

In general, the populations of energy levels are determined by including all processes

that both populate and de-populate any given level. These processes include de-

excitations from all higher levels and excitations from all lower levels – collisional,

radiative and spontaneous. In a steady state, the transition rate into any level equals the

rate out, and this should be true for all levels, a situation that is called statistical

equilibrium. Equations of statistical equilibrium are set up for each level and involve

the density of the particles, the energy density of the radiation field, and coefficients

describing collisional, radiative, and spontaneous transition probabilities. The coeffi-

cients may themselves be functions of other quantities such as quantum mechanical

parameters or temperature. The equations are then solved to determine the populations

of the levels for given conditions. This is a complex problem and is not always easily

solved. There are conditions, however, for which determining the population of states

can be simplified (see next section for a mathematical description). These are when the

gas is in thermodynamic equilibrium (TE) or in local thermodynamic equilibrium

(LTE).

If a gas is in TE, then the energy in the radiation field is in equilibrium with the

kinetic energy of the particles. Such a situation would be set up if a gas were


isolated and confined and no radiation escaped or entered the region. Eventually

the radiation and particles would achieve an equilibrium in which energy is

exchanged between photons and particles with no net energy gain or loss overall.

The radiation field can be characterized by a radiation temperature, TR, which

is, in general, a temperature that is used in an equation to calculate parameters of

the radiation field such as the specific intensity or flux density (Eqs. 4.1 and 4.2 are

examples). In TE, therefore, TR ¼ T (see also, Sect. 4.1). There is no additional

radiative source (which could make TR > T) and no ‘sink’ to let photons escape

(allowing TR < T). The photons that are produced in the gas (represented via TR)

therefore must be tightly coupled to the random motions of the particles

(represented by T). For bound states, this means that excitations and de-excitations

are collisionally dominated, that is, collision timescales are shorter than the

timescales associated with photon interactions or spontaneous de-excitations. It

is this property that leads to a simplification of the equations of statistical

equilibrium.

It is difficult to find an example in nature in which true TE holds since energy

will leave all objects eventually (the object will cool), even if very slowly. Probably

the best example is the 2.7 K cosmic microwave background radiation (Sect. 3.1).

This radiation filled the Universe in the past25 (no loss of gas or photons from

the Universe is possible) and the particles and radiation were in equilibrium.

Small variations in temperature are observed with position, however, as shown in

Figure 3.2.

A more common situation is one of LTE, that is, the gas has the properties of TE, but

only locally. A good example is one in which some radiation can escape or leak out of

the gas as, for example, in stars. The interiors of stars have densities that are high

enough for the radiation and matter to be considered ‘trapped together’ locally and are

therefore in equilibrium at any given location, yet there is a large scale temperature

gradient and a net flux of radiation outwards to the surface. Thus, the radiation

temperature is close to the kinetic temperature over some relevant size scale. Example

3.5 provides a quantification of this condition.

There are many examples in astrophysics for which LTE (or close to LTE) conditions

hold. In LTE, it is still true that the populations of states are collisionally driven, a

condition that is more likely to occur when spontaneous de-excitation timescales are

long, or the gas is dense and/or hot (see Eq. 3.19). From an astrophysical point of view,

the condition, TR T is very useful, not only because of the simplification of the

statistical equilibrium equations, but also because, by observing the radiation, we can

infer the kinetic temperature of the gas.Wewill see this more explicitly in Chapters 4, 8

and 9.

25The temperature at the time this occurred was much higher than 2.7 K but we see the lower temperature becauseof the expansion of the Universe (see Sect. 7.2 and the caption to Figure 3.2).


Example 3.5

Provide an argument showing that the interior of the Sun is in LTE.

We start by computing the mean free path of a photon in the Sun’s interior, for which we

assume that the gas is fully ionized and adopt a cross-section for interaction to be the

Thomson scattering cross-section (see Sect. D.1.1)26 for which s ¼ 6:65 1025 cm2 (Eq.

D.2). The mean density of the Sun (Table G.3) is r ¼ M=ð43 pR3Þ ¼ 1:4 g cm3. The

corresponding number density is then n ¼ r=mmH ¼ 1:4 1024 cm3, where we have

used m ¼ 0:61 for a completely ionized gas with Solar abundance (Sect. 3.4.2). Then (Eq.

3.18)l ¼ 1=ðn sÞ 1 cm with these values. How much does the temperature change over

this distance? The central temperature of the Sun is Tc ¼ 1:6 107 K and the surface

temperature is Ts ¼ 5781 K (Table G.3) so the global temperature gradient in the Sun is

rT ¼ ðTc TsÞR

¼ 1:6107

6:961010¼ 2:3 104 K cm1. Thus, the average temperature change is

only 2:3 104 K over the distance that a photon would travel before being reabsorbed.

Since a photon that is released will be absorbed again in a region that is essentially at the

same temperature, an LTE condition is satisfied – at least to an accuracy of 104 K. A

more accurate derivation would have to take into account the true temperature gradient

corresponding to the adopted density at each interior point, as well as regions that are not

fully ionized.

For stellar interiors, the fact that radiation is ‘locally trapped’ has implications for

the transport of energy out of stars. The energy generated in the core does not travel

straight to the surface to be radiated away. Rather, the photon will travel a mean free

path and then either be absorbed, followed by re-emission of another photon in a

random direction, or else be scattered into another random direction. Thus there are

many such absorptions, re-emissions, and scatterings as the radiation makes its way

from the core to the surface, a process called diffusion. As this diffusion proceeds, the

mean photon energy also degrades from g-ray or X-ray energies in the core to UV,

optical, or IR photons at the surface, depending on the type of star.

Although the photons near the surface are not the same ones as are originally

generated in the core, the process can nevertheless be thought of as a single photon

taking a random walk from the core to the surface27 (Figure 3.15) and a diffusion

timescale can be determined. Since the mean free path of a photon will vary with radius

within the star, wewill now considerl to be an effectivemean over all interior radii. For

a random walk, it can be shown that, after taking N random steps of unit length, the

straight-line distance travelled is onlyffiffiffiffiN

p. Therefore, for a photon whose mean free

path is l that takes a random walk towards the surface of a star of radius, R?, we have

26Absorption processes that occur in an ionized gas, such as discussed in Sect. 5.4.1, are ignored, but these willlower the mean free path even more than we calculate in this example.27We ignore regions of the star that may be convective, that is, regions in which the gas itself has bulk motion up ordown, carrying energy with it.


R? ¼ffiffiffiffiN

pl. If the interaction time for scattering or absorption is small in comparison to

the time of flight of the photon during each step, then the time for a photon to diffuse

from the core to the surface is the distance the photon actually travels divided by its

speed (c),

td ¼Nl

c¼ R?

l

2 l

c¼ R?

2

l cð3:22Þ

For the Sun, the diffusion time is of order, 107 yr when the duration of the interaction

itself is included in the calculation28 (Prob 3.7). This means that if there is a change in

the rate at which energy is generated in the core of the Sun, it would take about 107 yr

before we even knew about it29.

3.4.5 Excitation and the Boltzmann Equation

We now assume LTE and return to the issue of the population of energy levels in atoms.

For this condition, the equations of statistical equilibrium are much simplified and the

Random walk

of photon

from core

Net directionof travel to surface

Figure 3.15. In the interiors of stars, photons travel a mean free path before interacting with aparticle. The net result is a random walk to the surface.

28The duration of the interaction is only important for absorption/re-emission interactions; scattering should bealmost instantaneous.29As judged by the solar light output. Neutrinos, however, escape freely from the core. They are difficult to detectbut observatories like the one shown in Figure I.3.a could detect a change in neutrino output more quickly.


population of states is given by the Boltzmann Equation,

Nn

N1

¼ gn

g1eðDE

k TÞ ð3:23Þ

Here, Nn is the number of atoms in which electrons are in an energy level, n, and N1 is

the number of atoms with electrons in the ground state, where n is the principal

quantum number. DE is the energy difference between state, n, and the ground state

and gn is the statistical weight (see Appendix C.2). For hydrogen, gn ¼ 2 n2. The

Boltzmann equation can also be written to compare the populations between any two

states, rather than a comparison with the ground state only. If the Boltzmann Equation

holds, then LTE conditions are present.

The Boltzmann equation is similar to the Maxwell–Boltzmann velocity distribution

equation (Sect. 3.4.1), which should not be surprising given the discussion in the

previous section. The populations of states depends on the Boltzmann factor, ex=ðk TÞ,x representing the excitation energy, as well as a quantity that indicates the number of

possible states in which a particle could be found. Thus, Eq. (3.23) is like a discrete, or

quantum mechanical analogue to the Maxwellian velocity distribution.

There is a difficulty with Eq. (3.23) in that as n ! 1, gn ! 1 yet

eðDEkTÞ ! constant because, at high excitation, DE just becomes the ionization energy

of the atom. Thus the Boltzmann equation diverges at very high n. Physically, however,

before n reaches values at which the Boltzmann equation diverges, other effects

become important that effectively impose an upper limit to the principal quantum

number, nmax. For example, the electrons at high n may become stripped off by the

effects of neighbouring atoms. Coulomb interactions between the free charged parti-

cles and their distribution in the gas can also affect nmax.

Rather than considering how many particles are in an excited state in comparison to

the ground state, a more general form of the Boltzmann equation is one which

compares the number of particles in any excited state, Nn, to the total number of

neutral particles, N ¼ N1 þ N2 þ :::þ Nnmax . From Eq. (3.23),

Nn

N¼ gn e

ðDEnkT

Þ

g1 eðDE1

kTÞ þ g2 e

ðDE2k T

Þ þ :::þ gnmaxeðDEnmax

kTÞ¼ gn

UeðDE

k TÞ ð3:24Þ

where U Pnmax

n¼1 gn eðDEn

kTÞ is called the partition function.

At temperatures below 3500 K, the partition function for hydrogen reverts to the

statistical weight of the ground state, i.e. g1 eð 0

kTÞ ¼ g1 ¼ 2 ð1Þ2 ¼ 2 since the remain-

ing exponentials in the denominator are very small for low T. For hydrogen at higher

temperatures such as in the atmospheres of stars, the partition function has been

parameterized by,

logðUÞ ¼ 0:3013 0:00001 logð5040=TÞ ð3:25Þ


which is valid over the temperature range, 3500–20 000 K (Ref. [68])30. Evaluating this

equation shows that, again, U ¼ 2 to within 0.1 per cent over this temperature range.

Thus, for HI, Eq. (3.24) is essentially equivalent to Eq. (3.23) for temperatures up to

20 000 K. Example 3.6 shows that very high temperatures are required before there are

significant numbers of particles in even the first excited state. Such high temperatures,

as we shall see in the next section, will ionize the gas.

Example 3.6

Determine the temperature required for the number of particles in the first excited state ofhydrogen to be 10 per cent of the number in the ground state.

By the Boltzmann Equation (Eq. 3.23), the equation for statistical weight (Eq. C.11), and

DE (Eq. C.6) expressed in erg with n0 ¼ 2 and n ¼ 1 we have,

N2

N1¼ 2ð22Þ

2ð12Þ exp 2:181011ð 1

22 1

12Þ

k T

. Setting the LHS to 0.1 and solving, we obtain

T ¼ 3:2 104 K. Thus, the temperature must exceed 30 000 K in order for there to be a

significant number of particles in the first excited state.

When the gas density is low, such as in the ISM, LTE often no longer holds so the

Boltzmann Equation, as stated in Eq. (3.23), cannot be used. The rate at which an

electron will spontaneously de-excite is determined by the Einstein A coefficient

which, for de-excitation from the first excited state to the ground state (Ly a) is

A2;1 ¼ 6:3 108 s1 (Appendix C plus notes to Table C.1). The collision rate, how-

ever, is of order, 106 ne, where ne is the number density of free electrons, i.e. the

electron density (Ref. [160]). Any transition to the first excited state, whether radia-

tively or collisionally induced, will therefore result in an immediate spontaneous de-

excitation for the range of densities observed in interstellar space (Table 3.1). Similar

arguments hold for the other excited states of hydrogen (see Table C.1), indicating that

hydrogen in the ISM is not in LTE. Moreover, this implies that neutral hydrogen in the

ISM is in the ground state.

The Boltzmann Equation (Eq. 3.23) defines the excitation temperature, Tex. i.e. Tex

iswhatever temperature, when put into the Boltzmann Equation, results in the observed

ratio of Nn

N1(or whatever two levels are being compared). For a gas in LTE, all levels in

the atom can be described by the same value of Tex, in which case the excitation

temperature is just the gas kinetic temperature. In non-LTE cases, the Boltzmann

Equation may still be used provided the excitation temperature, rather than the kinetic

temperature, is inserted into Eq. (3.23). There could then be a different value of Tex

between every pair of energy levels in the atom. There are even cases in which Tex

could be a negative value if there are more particles in an excited state in comparison to

a lower state. A ‘negative temperature’ only has meaning in the context of this

30Pressure effects may increase the error at the highest temperatures.


equation. An example of such a situation is an astrophysical maser, as illustrated in

Figure 3.1631.

ISM HI clouds are particularly interesting because, although all particles are in the

ground state, this level is split into two hyperfine states due to the two possible

orientations of the proton and electron spins (see Appendix C.4). The excitation

temperature of this line is therefore called the spin temperature, TS. Since the sponta-

neous de-excitation rate of the upper hyperfine state is only A1;1 ¼ 3 1015 s1,

unlike the case for Ly a, spontaneous de-excitations are very rare and, for ISM

densities, this transition is collisionally induced. Since l21 cm radiation depends on

collisions with other neutral particles that have random motions described by the

kinetic temperature, for this transition, Tex T. Thus, we can say that Eq. (3.23)

31Maser stands for ‘microwave amplification by stimulated emission of radiation’. Masers are the microwaveversion of the more common ‘lasers’ (‘l’ for ‘light’) but the principle is the same, i.e. particles are pumped up to ahigher energy level that is ‘metastable’, meaning that the particle can remain in the higher state for a relativelylong time so that particles accumulate in the higher state. The particles then cascade downwards to a lower stateemitting a very strong signal at a single wavelength (coherent emission).

Figure 3.16. The nucleus of the edge-on spiral galaxy, NGC 3079 (Ref. [60]), harbours the mostluminous H2O maser known. The emission is at a rest frequency of 22 GHz, but the x axis in the insetspectrum (Ref. [172]) has been converted to a velocity via the Doppler formula (Table I.1). Theexcitation temperature of this line is negative since, at any time, there are more particles in theupper energy level in comparison to the lower energy level. In astronomical sources, masers areusually associated with molecules such as H2O or OH which are near sources of pumping such as hotstars or the nuclei of galaxies.


holds for this particular transition and the HI spin temperature is about equal to the

kinetic temperature of the gas. This is a very useful result that will allow us to compute

the temperatures of interstellar HI clouds from the l21 cm line (see Sect. 6.4.3).

3.4.6 Ionization and the Saha Equation

The ionization state of a gas in LTE can also be expressed in a fashion similar to the

Boltzmann Equation. For a gas of temperature, T, the number of particles that are in the

K þ 1 state of ionization compared to the number of particles that are in theKth state is

given by the Saha Equation,

NKþ1

NK

¼ 2UKþ1

ne UK

2pme k T

h2

3=2

exKk T ð3:26Þ

where UKþ1, UK are the partition functions of the K þ 1 and Kth ionization states,

respectively, ne is the electron density, me is the electron mass, xK is the energy

required to remove an electron from the ground state of theKth ionization state, and the

other quantities have their usual meanings. Like the Boltzmann Equation, higher

temperatures result in a higher excitation state – in this case, a higher fraction of

particles that are ionized. However, unlike the Boltzmann Equation, the Saha Equation

has a dependence on electron density. If there is a higher density of free electrons, then

there is a greater probability that an electron will recombine with the atom, lowering

the ionization state of the gas32. Regardless of which atom is being considered in the

Saha Equation, the electron density must include contributions of free electrons from

all atoms. Heavier atoms, for example, may contribute to ne in larger proportion than

their abundances (Figure 3.9) would suggest since they have more electrons and since

their ionization energies may be lower33.

Since hydrogen only has one electron to be removed and can only exist in the singly

ionized or neutral states, Eq. (3.26) reduces to,

NHII

NHI

¼ 2:41 1015T3=2

nee

1:58105

T ð3:27Þ

where we have used xHI ¼ 13:6 eV, UHI ¼ 2 (Sect. 3.4.5), and UHII ¼ 1 since the

ionized hydrogen atom is just a free proton and only exists in a single state. It is now

straightforward to show that a hydrogen atom tends to become ionized before there are

many particles in even the first excited state (Example 3.7).

32An exception is if the densities are so high that electron degenerate pressure (Sect. 3.3.2) starts to becomeimportant in which case the ionization state remains high even though ne is also high (Prob 3.8).33It is generally easier to strip off an outer electron from an atom that has other electrons around it since theeffective nuclear potential is weakened.


Example 3.7

At the surface of a hot star (Ref. [92]) the conditions are: T ¼ 26 729 K and ne ¼9:12 1011 cm3. Assuming that LTE holds, determine the fraction of all hydrogen atomsthat are ionized and compare this to the result of Example 3.6.

Using Eq. (3.27),

NHII

Ntot

¼ NHII

NHI þ NHII

¼NHII

NHI

1þ NHII

NHI

¼ 3:1 107

1þ 3:1 107 1

Example 3.6 showed that a temperature of 32 000 K was required to place 10 per cent of all

neutral hydrogen atoms in the first excited state. Yet, from this example, it is clear that at an

even lower temperature, virtually all hydrogen is ionized. Thus, a hydrogen gas starts to

become appreciably ionized at temperatures lower than those required to excite the neutral

atom. This is because there are many more possible states available for a free electron, than

for a bound electron in the first excited state.

We conclude that, if a hydrogen gas in LTE is neutral, virtually all particles will be in

the ground state. Since we have already seen that neutral hydrogen that is not in LTE

will also be in the ground state, our conclusion is even firmer.

3.4.7 Probing the gas

In order to probe the nature and properties of a gas, it is necessary to observe the gas

in some tracer that is appropriate to its ionization and excitation state. From the

previous sections, we know that hydrogen that is neutral will have its electrons in

the ground state and therefore no emission lines, such as from the Lyman, Balmer,

Paschen, etc. series (Appendix C) would be seen since few electrons are in excited

states. Thus, to obtain information on HI such as, for example, the shell shown in

Figure 3.17, we require some probe of the ground state itself. This is provided by the

l 21 cm line that is introduced in Appendix C and will be discussed further in Sect.

6.4.3. Another possibility is if a background signal excites the electron from the ground

state upwards. The upwards pumping will produce absorption lines since the back-

ground photon that would normally pass through the gas is removed from the line of

sight (Sect. 6.4.2).

Ionized hydrogen exists wherever there is a source of radiation or collisions

energetic enough to eject the electron from the H atom. Examples are HII regions

like the ones shown in Figures 3.13 and 8.4 in which an embedded hot star (or stars)

provides the radiation field, or planetary nebulae, as in Figure 3.7, in which the radiation

field is supplied by the central white dwarf. There are several ways to probe ionized

gas. One is to observe the emission that occurs when the free electrons in these nebulae


accelerate near positively charged nuclei (see Sect. 8.2). Another results from electrons

recombining with nuclei. Electrons always have some probability of recombining with

positively charged nuclei, so continuous ionizations and recombinations occur in a

steady state in an ionized gas. This means that emission lines of hydrogen such as the

Lyman and Balmer series lines can be observed as electrons cascade down various

energy levels, emitting photons in the process. Even though the gas is highly ionized,

these transitions between bound states in atoms (meaning the atom is momentarily

neutral) can be observed. Such emission lines are called recombination lines (see also

Figure C.1 and Sect. 9.4). Wherever such hydrogen emission lines are seen, the gas

from which these lines originate must be ionized.

Gases that are partially ionized are also present in a number of astrophysical

conditions such as stellar atmospheres or interiors, or diffuse interstellar regions farther

from sources of ionizing radiation than the immediate HII region. Such gases may also

be detected by their emission lines, depending on the degree of ionization and the

number of particles that are present. For stellar atmospheres, however, an important

Figure 3.17. An expanding 1:8 105 M HI shell in the low density outer region of the Milky Waygalaxy (distance from the Galactic Centre ¼ 23.6 kpc), observed in the l 21 cm line of neutralhydrogen. This shell is believed to represent the ISM HI that has been pushed outwards after asupernova exploded 4.3 Myr ago and after the emission from the supernova itself (e.g. similar tothat shown in Figure 1.2) has faded from view. The radius of the shell is 180 pc, the expansionvelocity is 11:8 km s1 and the temperature is 230 K. (Ref. [162]). Composed for the CanadianGalactic Plane Survey by Jayanne English with the support of A. R. Taylor. (Reproduced bypermission of J. English)


probe of the conditions is provided by absorption lines. Absorption lines are formed

when the temperature of the background stellar ‘surface’34 is hotter than the partially

transparent stellar atmosphere that surrounds it. The background radiation excites

the atmospheric atoms and absorption lines are seen, as described above (see also

Sect. 6.4.2). Historically, the most important hydrogen absorption lines are those of the

Balmer series since they occur in the optical part of the spectrum. However, observing a

Balmer absorption line requires that there be a sufficient number of hydrogen atoms in

which electrons are in the energy level, n¼ 2. As we know, few particles are in that

state, but a small fraction can still produce an observable line. By considering the

Boltzmann and Saha Equations together, it can be shown that the fraction, N2=Ntot,

reaches a maximum at a temperature around 104 K (see Figure 3.18). At lower

34The surface occurs at the radius at which the star becomes opaque. See comments in Sect. 6.4.2 and Footnote 12in Chapter 6. For Solar values, see Table G.11.

0

2e–07

4e–07

6e–07

8e–07

1e–06

1.2e–06

1.4e–06

0 2000 6000 10000 14000 18000Temperature (K)

))IIH(

N+)IH(

N(/2_N

Figure 3.18. Fraction of hydrogen atoms that are in the first excited state, N2, in comparison tothe total number of hydrogen atoms, NHI þ NHII . The adopted electron density at all temperatureshas been taken to be constant at ne ¼ 1013 cm3 (in reality, this will also vary with temperature).This distribution indicates how the strength of the Balmer absorption lines varies with stellarsurface temperature. Note that the fraction is still quite low, even at the peak.


temperatures, there are insufficient numbers of particles with electrons in n¼ 2, and at

higher temperatures, most of the particles have been ionized. Thus, the strength of the

Balmer absorption lines seen in stars reaches amaximum around 104 K. The strength of

these and other absorption lines in stellar atmospheres forms the basis of the stellar

classification system in use today. The properties of the various stellar spectral types,

which are ordered by temperature, are listed in Tables G.7, G.8, and G.9. Sample

spectra, showing how various absorption lines changewith spectral type, are illustrated

in Figure 6.8. Since stars are at 104 K for spectral type, A0V, this is the type for which

the Balmer lines are strongest.

3.5 The dusty Universe

Of order 2 106 kg of meteoritic dust (also called micrometeorites) in the mass

range, 105 to 102 mg descend upon the Earth each year (Ref. [104]). Analysis of

these particles and larger meteorites as well as data resulting from space explora-

tion using probes, imagers, and landers, have yielded a wealth of information about

the particulate matter in the Solar System. For the first time, we have now collected

dust from a comet and returned the particles to the Earth for study (Figure 3.19).

Yet, it is not clear to what extent local material resembles the interstellar dust that

is scattered throughout the Galaxy. This means that our knowledge of interstellar

medium (ISM) dust properties must still be deciphered from the light that is

emitted, absorbed, scattered and polarized by this important component of the

ISM. In this section, we will look at some of the observational effects of dust and

discuss dust properties. Details of the interaction of light with dust will be dealt

with in Sect. 5.5.

Figure 3.19. The impact of a cometary dust grain onto aerogel (a sponge-like silicon-based solidthat is 99.8 per cent air) is shown here, captured when NASA’s Stardust spacecraft flew through thedust and gas cloud surrounding Comet Wild 2 on January 2, 2004. Stardust returned to Earth onJanuary 15, 2006. Inset: A tiny particle collected by the Stardust spacecraft. This particle is 2 mmacross and is made of a silicate mineral called forsterite, known as peridot in its gem form.(Reproduced by permission of NASA/JPL-Caltech and the Stardust Outreach Team)


3.5.1 Observational effects of dust

Almost everywhere we look in the sky, we see some evidence for dust. Dark bands

that cut a swath across galaxies (Figure 3.22), thin curves that follow spiral arms

(Figure 3.8), filamentary patterns in HII regions (Figure 3.13), patchy dimming of

starlight in the Milky Way (Figure 3.20), diffuse glows from scattered light around

individual stars (Figure 5.3), and stellar disks hinting of hidden planetary bodies

(Figure 4.13) all attest to the presence of small particulate (solid) matter in a wide

variety of environments.

Although dust makes up only 1 per cent of the ISM, by mass, its effects are

extremely important, especially in the optical and UV where dust strongly scatters and

absorbs, obscuring the light from behind. Dust prevents us from seeing the nucleus of

ourMilkyWay galaxy (Figure 3.3) at optical wavelengths and wemust rely, instead, on

other wavebands to penetrate through this barrier. Indeed, the depth of our view along

the plane of the Milky way is quite shallow, being restricted to about a kpc, on average,

though there are some lines of sight that are clearer. What is worse, the distribution of

dust is so patchy that it cannot be easily modelled or corrected for, except on an object-

by-object or location-by-location basis. In the history of our understanding of the

Milky Way, dust has been a major impediment to obtaining distances and sizes of

various objects, including the large-scale structure of the Galaxy itself. For example, if

the absolute magnitude, M, is believed to be known for some object, say via a

calibration of other similar objects of known distance, then a measurement of the

Figure 3.20. Left: At one time, dark regions like this in the sky were thought to be ‘holes’ –regions in which there were no stars. We now realize that these dark nebulae are dense molecularclouds. The dust makes up only about 1 per cent of the mass of the cloud yet it is the dust thatobscures the background starlight from view. This cloud, called Barnard 68, is quite nearby, about150 pc away and 0:15 pc across. Right: The same cloud is shown by the emission of a spectral line ofthe carbon monoxide isotope, C18O. Most of the mass of the cloud is in molecular hydrogen, H2, butvarious other molecules and their isotopes are present, such as CS, N2H

þ, NH3, H2CO, C3H2, amongothers. (Lada, C. U., Bergin, E. A., Alves, J. F., and Huard, T. L., 2003, ApJ, 586, 286. Reproduced

by permission of the AAS and J. Alves)

3.5 THE DUSTY UNIVERSE 105

apparent magnitude, m, leads to a measurement of distance via Eq. (1.30). If there is

dust along a line of sight, however, the object will look dimmer than it otherwise would

be, leading to a distance that is erroneously large. Early estimates of the size of the

MilkyWay were high by at least a factor of 2 for just this reason. In the presence of dust

obscuration, therefore, Eq. (1.30) must be re-written to take this into account,

VMV AV ¼ 5 log d 5 ð3:28Þ

The quantity, AV , is the total extinction or just extinction35 in the V band which gives

the amount of dimming, in magnitudes, as a result of scattering and absorption along

the line of sight, that is (Eq. 1.26),

AV ¼ 2:5 logfV

fV0

ð3:29Þ

where fV is the flux density of the object with extinction and fV0 is the flux density in the

absence of extinction. Similar equations could be written for the other filter bands.

Since the V band extinction per unit distance along the line of sight in the plane is about

1 mag kpc1 (Ref. [180]) large errors in distance will occur if the extinction is ignored.

For example, an object at a true distance of 1 kpc will be placed at a distance of 1:6 kpcif dust extinction is not included (Prob. 3.11).

Reddening was introduced in Sect. 2.3.5 in the context of the Earth’s atmosphere.

Outside of the Earth’s atmosphere, the main cause of reddening at optical wavelengths

is dust. Reddening is therefore seen in the interstellar medium or in any dusty

astrophysical environment. It is due to the fact that dust extinction is more effective

at short wavelengths than at long wavelengths, so any object viewed at optical

wavelengths will appear redder if seen through a veil of dust. Given this wavelength

dependence of extinction, we can define a selective extinction, which is a measure of

reddening at some wavelength, l,

ElV ¼ Al AV ð3:30Þ

It is possible to measure the reddening of an astronomical object via a comparison with

another object (a calibrator) that has negligible reddening. For example, if two stars of

the same intrinsic type, and therefore the same absolute magnitude, are observed in the

V and B bands, the apparent magnitudes of the star of interest and the calibrator,

respectively, will be,

V ¼ MV þ 5 logðdÞ þ AV ðstarÞB ¼ MB þ 5 logðdÞ þ AB ðstarÞ ð3:31ÞV0 ¼ MV þ 5 logðd0Þ ðcalibratorÞB0 ¼ MB þ 5 logðd0Þ ðcalibratorÞ ð3:32Þ

35‘Extinction’ is also used as a general term that refers to the combined effects of scattering and absorption (seeChapter 5).


where d and d0 are the distances to the star of interest and the calibrator, respectively.

Combining and rearranging these equations, and expressing ðB0 V0Þ as ðB VÞ0,yields,

EBV ¼ AB AV ¼ ðB VÞ ðB VÞ0 ð3:33Þ

This is an important result because it does not depend on knowing distances which

are often difficult to obtain. One need only compare the quantities, B V, for the

star of interest and the calibrator star. Differences of magnitudes are also more

accurately determined than the magnitudes themselves, in the event that there is

some error in the absolute scale being used. It is straightforward, then, to determine

ElV for any l, provided unreddened (or minimally reddened) calibrators can be

found. Note that EBV is almost always a positive quantity since blue light is more

heavily extincted (will have a higher numerical value of the magnitude) than visual

light. Thus, by Eq. (3.33), we have a way of determining the selective extinction of

an object.

It is less straightforward to determine the total extinction, AV. However, much effort

has gone into determining this value for objects of known distance. As a result, it has

been found that the ratio of total to selective extinction is very nearly constant in the

Milky Way. This is basically a statement that the same dust that is producing the total

extinction is also producing the reddening. The two values are related via (Ref. [180]),

RV ¼ AV

EBV

¼ 3:05 0:15 ð3:34Þ

In the absence of other information, if the selective extinction is measured, this relation

may be used to find the total extinction which can then be used in Eq. (3.28), as required

(Example 3.8).

Example 3.8

An M0III star is measured to have blue and visual magnitudes of B ¼ 8:76 and V ¼ 7:09.Estimate the distance to this star.

From Table G.8, ðB VÞ0 ¼ 1:57 for a M0III star. Therefore, the selective extinction of

this star is (Eq. 3.33) EBV ¼ ðB VÞ ðB VÞ0 ¼ ð8:76 7:09Þ 1:57 ¼ 1:671:57 ¼ 0:10. From Eq. (3.34), AV ¼ 0:305. Using MV ¼ 0:2 for this kind of star

(Table G.8), Eq. (3.28) yields a distance of 249 pc.

Historically, since most observational attention has been paid to the optical wave-

band, the optical extinction properties of dust have dominated the attention of astron-

omers for the purposes of distance or flux corrections. In other words, the dust has been

considered ‘in the way’. However, for dust, it is certainly true that ‘one man’s trash is


another man’s treasure’. The same dust that blocks our optical view of the cosmos is a

crucial component in the complex chemistry of the ISM, it traces magnetic fields, plays

an important role in ISM dynamics, must be taken into account in abundance determi-

nations, and contributes to driving stellar evolution. Thus the study of dust is a rich and

important sub-field of astronomy in its own right.

Observational data that allow us to probe the characteristics of dust are therefore

very important. An example is Figure 3.21 which shows the mean extinction curve for

the Milky Way in the optical band. The extinction curve is a plot of the selective

extinction, ElV (normalized by EBV) as a function of l1. Since ElV ¼ Al AV,

this plot shows how the extinction varies with wavelength. In the optical region, a rough

approximation is that ElV / 1=l.Structure in the extinction curve provides clues as to the make-up of interstellar dust.

For example, there are several changes in slope and an obvious peak (excess extinction)

in the ultraviolet at 4:6 mm1 (l2175A). Such spectral features can be compared to

laboratory spectra of vibrational modes within solids in an attempt to determine the

type of material responsible. For the l2175A peak, the best candidate is graphite

(Figure 3.23) or a related particle (Ref. [180]). See Example 5.8 for further details on

the link between the extinction curve and the optical properties of grains.

In reality, because a complete description of dust requires many parameters, it is

difficult to find a unique result using only the extinction curve. This is compounded by

the fact that there may be different types and admixtures of grains in different

Figure 3.21. The mean extinction curve for the ISM in the optical band from 0:2 ! 10mm1

(l5 000 ! l100 nm). Frequency increases to the right and the V and B bands are marked. Note thatlarger ElV

EBVcorresponds to more extinction, so background interstellar light will be more heavily

extinguished at these wavelengths. (Whittet, D.C.B., Dust in the Galactic Environment, 2nd Ed.,Institute of Physics Publishing)


interstellar environments. A good example is our own Solar System in which solid

particles (and planets) become icier farther from the Sun. Fortunately, there are other

ways of obtaining additional information about interstellar dust. These include the

study of dust scattering and polarization characteristics (Sect. 5.5). In addition, a rich

area of study is now the emission properties of grains viewed in the infrared part of the

spectrum. The same dust that causes extinction in the optical band will radiate in the IR

(see Sect. 4.2.1). This is beautifully illustrated by Figure 3.22. With the advent of

infrared telescopes such as the Infrared Astronomical Satellite (IRAS) launched in the

1980s, the Spitzer Space Telescope, launched in 2003, and the most powerful of these,

the Herschel Telescope, the study of dust has become a growing and maturing sub-

discipline in astronomy. Put together, we have an emerging view of dust characteristics,

as described in the next sections.

3.5.2 Structure and composition of dust

Large grains, also called classical grains, are of order, 0:1 mm in size which is about

the size of the particles in smoke. Very small grains (VSGs) are of order, 0:01 mm

Figure 3.22. The Sombrero Galaxy (M 104) in optical and infrared light. (a) A Hubble SpaceTelescope image reconstructed from a series of images taken in red, green and blue filters. The dustlane obscures the light from the stars behind it. (Credit: Hubble Space Telescope/Hubble HeritageTeam) (b) False colour image showing l3:6mm data in blue, l4:5 mm data in green, l5:8 mm data inorange, and l8:0mm data in red. In this image, blue/green represents starlight primarily from starscooler than the Sun, while red/orange represents thermal emission from dust at its equilibriumtemperature. Now the dust lane is emitting, rather than obscuring, and starlight can be seenthrough it. [Reproduced by permission of NASA/JPL-Caltech/R. Kennicutt (University of Arizona),and the SINGS Team] (See colour plate)


which is about the size of a small virus. However, a continuum of particle sizes is likely

present in a power law distribution such that there are more smaller grains and fewer

large grains, i.e. nðaÞ / aq, where nðaÞ is the number density per unit size range

(cm3 cm1) of grains of radius, a, and q is the power law index that must be fitted to

the data. Ignoring some complexities, a function that fits the data reasonably well over

the range, a ¼ 0:005mm to a ¼ 0:25mm is (Ref. [180]),

nðaÞ / a3:5 ð3:35Þ

Grain composition is more difficult to ascertain since few spectral features are

observed from these solid particles. Carbon appears to play an important role and,

given its propensity to join together with other carbon atoms, this element can

combine in a wide variety of ways, including highly ordered (crystalline) struc-

tures such as nanodiamonds, graphite, and fullerenes (see Figure 3.23) as well as

amorphous (irregular) structures. Carbon can also easily form hydrogenated rings,

called polycyclic aromatic hydrocarbons (PAHs, Figure 3.24), commonly seen on

Earth in the soot from automobiles. Some version of PAHs, though possibly not

exactly in the same formations, may be in the interstellar medium, as suggested by

observed mid-infrared spectral features. PAHs are planar, rather than three dimen-

sional, and so are considered to be large molecules, containing of order 50 atoms,

rather than the three-dimensional structures that we identify with solid dust grains.

PAHs are therefore at the transition between molecules and dust. Another compo-

nent of dust is likely silicates, a class covering a wide variety of chemical

compositions involving SiO4. These appear to be in amorphous, rather than

crystalline, structures and also likely contain magnesium (Mg) and iron (Fe). A

third category is ices, such as the solid phases of water (H2 O), methane (CH4) and

ammonia (NH3).

The structure of the grains is less certain. ‘Sooty’ (carbon rich) mantles over silicate

cores is one possibility. Ices may also form mantles about cores since they move more

easily between the gas phase and the solid phase. As Figure 3.23 illustrates, dust grains

are unlikely to be uniform spheres. It is clear that at least some fraction of grains are

elongated, since this explains the fact that scattering from dust results in polarized

light, as shown in Figure D.6.

Figure 3.23. Sketch of the three different ordered forms in which carbon can exist in its solidphase. (Whittet, D. C. B., Dust in the Galactic Environment, 2nd Ed., Institute of Physics Publishing)


3.5.3 The origin of dust

Where do these grains come from? It is well known that the envelopes of stars contain

dust. The cool atmospheres of evolved stars appear to be a source of dust since their

atmospheres are rich in carbon and oxygen (Sect. 3.3.2) and dust can easily be driven

from the extended envelopes of these stars into the ISM via radiation pressure (Sect.

1.5). However, it appears that ‘stardust’, although important, cannot be the sole source

of interstellar dust, since more dust is present in the ISM than can be accounted for by

stellar injection alone (Ref. [180]). An additional source of dust now appears to be the

metal-rich ejecta of supernovae. For example, the supernova remnant, Cas A, shown in

Figure 1.2, contains 2–4 M of dust, which is about 100 times the amount of dust that

could have been swept up from the ISM during its expansion. This implies that dust was

formed during the supernova event, itself (Ref. [54]).

Observationally, we know that the dust distribution in the Galaxy follows the dense,

molecular gas distribution (Figure 3.20). There appear to be complex processes in these

regions involving gas-phase chemical reactions, the absorption of particles onto grains,

migration and reactions on grain surfaces, destruction of grains in shock waves36, and/

or growth of grains via coagulation or deposition onto mantles. These processes are not

yet fully understood, but the careful study of light associated with dust-rich regions is

slowly revealing the properties and nature of this important component of our Uni-

verse. The intimate link between dust grains and large molecules, especially organic

molecules and carbon-based molecules such as PAHs, and related information such as

Pyrene C16H10

Coronene C24H12

Perylene C20H12

Benzo[ghi]perylene C22H12

Anthanthrene C22H12

Ovalene C32H14

Figure 3.24. Sample configurations of polycyclic aromatic hydrocarbons (PAHs) which may make upthe largest molecules in the ISM. (Reproduced using data from http://pubchem.ncbi.nlm.nih.gov/)

36See Footnote 16 in this chapter.


the presence of amino acids in meteorites, hint that this branch of astrophysics may

provide important clues about the origin of life itself.

3.6 Cosmic rays

The existence of cosmic rays (CRs) has been known since 1912 when Victor Hess took

an electrometer (a device for measuring the presence of charge) to a high altitude in a

hot air balloon. Contrary to what was expected, the electrometer discharged more

quickly at higher altitudes, leading Hess to conclude that the source of the discharge

must be from above the atmosphere rather than the Earth itself. For this discovery,

he shared the 1936 Nobel Prize for physics. Called cosmic rays because they were

originally thought to be part of the electromagnetic spectrum, we know today that these

are high energy particles coming from space. As noted in the Introduction, CRs,

together with meteorites and meteoritic dust (see last section) represent the only known

particulatematter reaching the Earth’s surface. As such, they are important carriers of

cosmic information beyond that brought to us by light.

Figure I.1.a shows an early photograph of a cloud chamber containing CR tracks.

One need only watch a real cloud chamber for a few seconds to see this subatomic

world come to life. What is being viewed, however, are not the primary cosmic rays

that impinge upon the Earth’s atmosphere from above, but rather secondary particles

that are created within the atmosphere from cosmic ray showers. These occur when a

primary cosmic ray collides with an atmospheric nucleus, shattering it and producing

secondary particles, some of which will decay and some of which may collide with

other atmospheric particles, creating a chain of subsequent events. Various subatomic

particles are formed in the process, such as pions, muons, and neutrinos. Secondary

cosmic rays are responsible for approximately one third of the naturally occurring

radioactivity on Earth.

Since we cannot detect primary particles from our location on the Earth’s surface, to

understand the origin of cosmic rays, it is necessary either to make corrections for

atmospheric interactions or to make measurements from space. In the following

sections, we assume that such corrections have been made and discuss those cosmic

rays that impinge on the Earth’s atmosphere.

3.6.1 Cosmic ray composition

What is the composition of cosmic rays? Approximately 98 per cent of these particles

are nucleons (protons and neutrons in nuclei without their electrons, see Sect. 3.3.3)

and the remaining 2 per cent are electrons and their positive counterparts, positrons. Of

the particles in the energy range, 108 1010 eV, 87 per cent are hydrogen nuclei

(protons), 12 per cent are helium nuclei (also called alpha particles), and 1 percent

are heavier nuclei (Ref. [147]). Compared to the elemental abundances of the Solar

System and interstellar medium shown in Figure 3.9, CRs are slightly underabundant in


hydrogen (we expect just over 90 per cent, Sect. 3.3) and overabundant, by several

orders of magnitude, in the light elements, Li, Be, and B. Moreover, CRs are highly

underabundant in electrons since, for a normal plasma, there should be approximately

equal numbers of protons and electrons. Thus, the admixture of CR particles is quite

different from what we normally see in the Solar System or ISM.

Most compositional anomalies can be explained by models that take into account the

interactions of CRs en route to Earth. For example, the over-abundance of light

elements can be understood in terms of spallation – the breaking up of heavier nuclei

into smaller particles via collisions (see also Sect. 3.3.4). It is clear that the ISM acts

like another ‘atmosphere’ which, as before, must be corrected for in order to under-

stand the original composition of these particles at the location where they are first

accelerated37. Few abundance anomalies remain once the ‘ISM atmosphere’ is taken

into account.

Some departures from Solar abundance do remain, however, and these can provide

clues as to the origin and acceleration mechanism of the particles (see also Sect. 3.6.3

below). One is the relative underabundance of hydrogen. Supernova ejecta, for

example, consist of mostly heavier particles (Sect. 3.3.3) in comparison to hydrogen.

Also, acceleration mechanisms may be more effective for interstellar grains (which

contain heavy elements) than for particles that are in the gas phase such as hydrogen

(Ref. [57]). Another is the element, 22Ne which is in excess compared to the Solar

value. This element is known to be rich in the spectra of Wolf–Rayet stars (very

massive, young hot stars, see also Sect. 5.1.2.2 and Figure 5.10), hinting at a possible

connection between CRs and massive stars that are rich in heavy metals.

The strong underabundance of electrons compared to nucleons can be understood in

terms of the relative energies and momenta of the particles, a subject to which we will

return in the next section.

3.6.2 The cosmic ray energy spectrum

Peculiar to CRs are relativistic speeds and consequent high energies. The total energy

of a particle whose rest mass is m0, moving at a speed, v, is,

E ¼ m0 c2 þ T ¼ gm0 c

2 ð3:36Þ

where T is the kinetic energy, m0 c2 is the rest mass energy, and g (from Table I.1) is

called the Lorentz factor, defined as,

g 1ffiffiffiffiffiffiffiffiffiffiffiffi1 v2

c2

q ð3:37Þ

37Because of this, ‘primary cosmic ray’ is sometimes used to represent the CRs near the acceleration site ratherthan those that hit the Earth’s atmosphere.

3.6 COSMIC RAYS 113

If v c, a binomial expansion (Eq. A.2) can be used on Eq. (3.37) to find,

g 1þ 1

2

v2

c2ð3:38Þ

In this case, Eq. (3.36) reduces to,

E ¼ m0 c2 þ 1

2m0 v

2 ðv cÞ ð3:39Þ

which is a more familiar expression showing both the rest mass energy of the

particle (first term) and its kinetic energy (second term). The electron and proton

rest mass energies are 0:51 MeV and 938 MeV, respectively, corresponding to g ¼ 1.

Electrons with energies greater than about 0:5 MeV or protons with energies

greater than about 1 GeV will therefore be dominated by their kinetic energies. If

we consider a ‘relativistic speed’ to be at least 0:3 c, then any particle with g > 1:05 isrelativistic. We will see, below, that Lorentz factors as high as g 1011 have now

been measured for some high energy CRs, corresponding to v ¼ c to many decimal

places.

The cosmic ray energy spectrum is shown in Figure 3.25 (over this range, E T),

showing a beautiful power law spectrum, with several changes in slope, over 13 orders

of magnitude in particle energy! Since this is a log–log plot, a straight line represents a

power law distribution of energies for these particles which can be described by,

JðEÞ ¼ K EG ð3:40Þ

where G is the cosmic ray energy spectral index, E is the energy of a particle and K

is a constant of proportionality. The quantity, JðEÞ, in units of particles

s1 m2 GeV1 sr1, is a kind of ‘specific intensity38 for particles’ (compare this

to a specific intensity for radiation which has units of erg s1 cm2 Hz1 sr1) and

can be integrated to obtain the total number of CRs hitting an object per unit

time (Example 3.9). A single value of G, as shown by the dashed curve in

Figure 3.25, does not fit the data perfectly. There are changes in the slope, two of

which are marked ‘the knee’ and ‘the ankle’ at energies of 4 1015 eV and 5 1018

eV, respectively, the value of the ankle being less certain. Except for the very

lowest and very highest energies, G 2:7 is a good fit to the data at energies lower

than the knee and G 3:0 fits well above it. A value of K for the upper end of the

spectrum is provided in Eq. (3.41). See Prob. 3.13 for a value of K at energies below

this. Changes in the slope have been scrutinized in some detail because it is thought that

they may represent changes in the origin and/or acceleration mechanism of the

particles.

38Usually ‘particles’ is left out of the units.


Example 3.9

Between the energies of 1016 and 1019 eV (above the knee), the spectrum of Figure 3.25 canbe fitted by the function,

JðEÞ ¼ 2:1 107 E3:08 s1m2sr1GeV1 ð3:41Þ

where E is in units of GeV (Ref. [101]).What is the total rate of cosmic rays hitting the

Earth’s atmosphere in this energy range?

We first integrate over the energy range, E1 ¼ 107 GeV to E2 ¼ 1010 GeV,

Z E2

E1

JðEÞ dE ¼ 2:1 107

2:08ðE2

2:08 E12:08Þ ¼ 2:78 108 s1 m2 sr1 ð3:42Þ

104

102

10–1

10–4

10–7

10–10

10–13

10–16

10–19

10–22

10–25

10–28

109 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021

Energy (eV)

J(E

) (m

2 sr

s G

ev)–1

Ankle

Knee

Figure 3.25. Cosmic ray spectrum for particles from 108 to 1021 eV (Ref. [20]). The ordinaterepresents JðEÞ (Eq. 3.40) and the abscissa represents the kinetic energy per particle (Eq. 3.36) forwhich E T over almost the entire energy range plotted). The dashed line shows a power law fitwith a single spectral index. Subtle changes in the spectral slope can be seen in two places: the‘knee’ (at E ¼ 4 1015 eV) and, to a lesser extent, the ‘ankle’ (E ¼ 5 1018 eV). (Reproduced bypermission of Simon Swordy)

3.6 COSMIC RAYS 115

Assuming that CRs are received from all forward angles in the direction of space but not

from rearward angles in the direction of the Earth, the integration over solid angle amounts

to a multiplication by p steradians (this is equivalent to the geometry of Example 1.2b but

for the receiving, rather than the emitting, case). The result is a flux of f ¼ 8:73108 s1 m2 at any position at the top of the atmosphere. Finally we integrate over the

surface area of the Earth at the top of the atmosphere, which we take to be 4pR 2, since

the radius of the Earth plus its atmosphere is approximately equal to R (6:371 106 m,

Table G.3). The result is 45 million cosmic ray particles s1 for this energy range.

The cosmic ray energy spectrum for electrons is also a power law but it shows some

differences with respect to nucleons, especially at low energies because electrons are

more strongly influenced by the Solar wind (see next section). It is likely that the

electron spectrum is only free of this modulation at electron energies above 10 GeV

(Ref. [101]). If the solar modulation is corrected for, the electron spectrum can be fitted

by the function (Ref. [34]),

JeðEÞ ¼ 412E3:44 ð3GeV < E < 2TeVÞ ð3:43Þ

whereE is the electron energy inGeVand the units of JeðEÞ are the same as in Eq. (3.41).

The value of the constant of proportionality is less certain than that of the slope, due to

difficulties in combining data from many different experiments using different instru-

ments. Thus, Eq. (3.43) indicates that the CR electron spectrum is significantly steeper

than the G 2:7 found for protons and heavier nuclei (mainly protons).

This difference in spectral index between electrons and protons can be accounted for

by the greater susceptibility of electrons to interactions and energy losses in the ISM. A

variety of energy loss mechanisms exist for both electrons and protons, but electrons

lose energymore readily via synchrotron radiation (see Sect. 8.5) and inverse Compton

scattering (Sect. 8.6) which do not affect nucleons. Since energy losses are greater

for higher energy particles, the result is a steepening of the CR spectrum. In fact,

the spectral indexes, G0, of both electrons and protons are believed to be the same

initially and in the range, 2.19G092.5, depending on the specific acceleration

mechanism (Ref. [15]).

We can now return to the problem of the observed low electron fraction of cosmic

rays, that is, a measured electron to proton number fraction of about 2 per cent. This

value is measured at fixed kinetic energy, T . Although a variety of acceleration

mechanisms may be present, several of the better understood mechanisms in the

intermediate energy range (e.g. shock acceleration, see next section) give a distribution

in momentum, p, which is the same for electrons and protons except for a constant of

proportionality, i.e. NðpÞ / pG0 , and assume that the total number of particles above

a fixed lower kinetic energy, T0, is the same for both electrons and protons. However, at

a fixed lower T0, the electron momentum is less than the proton momentum which

means that the electron distribution is being sampled to lower momenta where there are

more particles. For the total number of electrons to equal the total number of protons,


the number of electrons must be less than the number of protons at fixed T . Essentially

the plot of NðTÞ for protons is shifted to the right (higher energy) in comparison to the

electron spectrum. This argument is described mathematically in the Appendix at the

end of this chapter. See also Ref. [147] for further details.

3.6.3 The origin of cosmic rays

What kinds of astronomical sources can accelerate particles to such high energies? For

some of the energy range of Figure 3.25, the answer to this question is unknown and the

subject of much current research. To accelerate CR particles requires a source

or sources of very high energy, possibilities including shock waves associated with

supernovae, massive stellar winds, neutron stars, regions around black holes, active

galactic nuclei, g-ray bursts (Sect. 5.1.2.2), and others. Even when the most powerful

of sources are considered, accelerating particles to Lorentz factors of order 1011

(E 1020 eV) strains the limits of known acceleration mechanisms. Aside from the

problems of corrections for the atmosphere, ISM interactions and subsequent energy

losses with spectral steepening, there is an additional problem in simply identifying the

locations of the sources. Cosmic rays easily scatter from magnetic field lines in the

galaxy (except at very high energy, see below) and therefore propagate by diffusion,

similar to the way a photon would take a random walk out of the Sun (Sect. 3.4.4).

Therefore, even if the angle at which a particle impinges on the Earth can be

determined, it will not represent the true direction of the source. There are some

energy ranges, however, for which the origin of CRs is thought to be understood.

Below about 1 GeV, for example, the CR spectral index changes in Figure 3.25,

becoming much flatter. The arrival of these low energy cosmic rays is correlated with

Solar activity, indicating that these particles originate from the Sun. A stream of high

energy particles called the Solar wind continuously leaves the Sun, beginning in the

hot, tenuous Solar corona that is shown in Figure 6.6. In addition, bursts of high energy

particles also result from activity such as Solar flares (Sect. 6.4.2). Thus, we know that

at least one star is responsible for a fraction of the CR flux.

At higher energies, CRs are believed to come from outside of the Solar System

with the bulk of the particles originating from sources within the Milky Way.

Energetically, ordinary stars or isolated neutron stars in the Milky Way cannot

account for the total flux of these higher energy CRs. Supernovae, however, provide

sufficient energy. Only 10 per cent of the kinetic energy of all supernovae in the

Galaxy would be needed to explain the CRs in the Galaxy at energies up to about the

knee of Figure 3.25. This connection has been strengthened by theoretical success at

explaining the CR energies and spectral index via acceleration in the shock waves

created by supernovae. The detection of synchrotron radiation (see Sect. 8.5), which

is emitted by the electron component of CRs, in supernova remnants in the Milky

Way provides a further link, as does the detection of TeV g-rays from supernova

remnants. A supernova origin also connects CRs with hot, massive metal-enriched

stars since these are the only kinds of stars that produce supernovae (Sect. 3.3.3) and

3.6 COSMIC RAYS 117

can also account for the underabundance of hydrogen (Sect. 3.6.1). All evidence

considered, supernovae appear to be the dominant source of mid-energy cosmic rays

up to the knee.

Once energies above the knee are considered, the situation becomes murkier and the

origin of these CRs is not as well understood. At these energies, cosmic rays more

readily escape from the galaxy and the admixture of Galactic sources may also change

from one in which supernovae dominate. There is also the possibility that this point

represents a transition from Galactic to extra-galactic sources. CRs above the ankle are

even more mysterious, posing a challenge for modern theory. The fact that particles

with energies greater than 1020 eV are observed at all is a puzzle of great current

interest. If these particles are extra-galactic in origin, they will meet photons from the

cosmic microwave background (CMB, Sect. 3.1) while travelling through intergalactic

space. Interactions of such high energy particles, mostly protons, with CMB photons

can occur via a variety of reactions but, most notably, the formation of pions with a

subsequent significant energy loss for the protons. The energy at which this occurs,

about 5 1019 eV, is called the GZK (for Greisen–Zatsepin–Kuz’min) cutoff. This

puts a limit on the region of space within which higher energy particles could have

originated, a distance that is approximately 50Mpc (a few tens to a hundred, depending

on energy). A challenge for observers is also that the flux of these particles is extremely

low, only about 1 particle per square kilometre per century! On the positive side, at

such high energies, these CRs do not scatter so strongly off of magnetic field lines and

therefore their apparent angle in the sky should point backmore closely to their place of

origin. Pinning down the energy source for these highest energy CRs is therefore only a

matter of time.

Problems

3.1 (a) Write a single equation for the estimated lifetime of a star on the main

sequence, making the same assumptions as in Example 3.1. Evaluate known quantities

and express the result in years with the stellar mass and luminosity in units of M and L,respectively.

(b) Refer to Table G.7 and find the bolometric magnitude and luminosity of a 17M star.

Evaluate the main sequence lifetime of this star, assuming that the formation time of a star

and all subsequent evolution off the main sequence are small fractions of the main sequence

lifetime. If the star formation is continuous, determine how many generations of 17Mstars could occur during a single Solar lifetime.

(c) Evaluate the main sequence lifetime of a 0:34M star with the same assumptions as

in part (b) and compare it to the age of the Universe.

3.2 Approximating stellar mass distributions by the IMFs shown in Figure 3.10 and

provided in Eqs. (3.5) and (3.6):


(a) What is the the number density of all stars formed from 0.1 to 100 M in the disk and

in the halo?

(b) Determine the mean separation between stars in both regions of the Galaxy.

(c) Express the mean separations of part (b) in terms of a ‘typical’ stellar diameter,Dt, for

each region.

(d) Assume that the effective diameter for star–star collisions is 10Dt. If the random

velocities of stars in the disk are of order 10 km s1, compute the mean time between star–

star collisions in the disk. What can you conclude about the probability of such collisions in

the Galaxy, excluding stars in dense clusters (see Footnote 6 in Chapter 10).

3.3 For any given collection of stars that are formed in the disk of the Milky Way,

determine the fraction of stars that will become supernovae. If star formation continues in

the same region until the gas is used up, should the number of supernovae in comparison to

the total number of stars in the region, increase, decrease, or stay the same with time?

Explain.

3.4 From the Maxwellian distribution (Eq. 0.A.2), derive Eqs. (3.8) and (3.9).

3.5 Show that Eqs. (3.13) result from Eq. (3.1) and Eq. (3.12).

3.6 For the HI shell shown in Figure 3.17, assume that the initial ISM had Solar

abundance, uniform density, and T ¼ 200 K before the supernova swept up the gas.

(a) Calculate the average density, n, of particles in the pre-supernova (pre-SN) ISM. (You

may assume that the mean atomic weight of the metals can be approximated by that of

oxygen.)

(b) Determine, for the pre-SN ISM, l (AU), t (yr), R (s1), and r (cm). Is this an ideal

gas?

3.7 (a) Calculate the time it would take a photon to travel from the core to the surface of

the Sun if there were no interactions en route.

(b) Adopting a mean free path of l ¼ 0:3 cm, determine the time (years) it takes for a

photon to diffuse from the core to the surface of the Sun (see Eq. 3.22) when the interaction

time is negligible.

(c) Write an expression for the diffusion time of a photon assuming each interaction

takes 108 s and re-evaluate this time for the Sun.

3.8 For the following two ionized regions, determine the mean separation between

particles, r, and the mean free path for a photon, l. Indicate whether the region should

be considered an ideal gas and also whether it is in LTE. For simplicity, assume that

3.6 COSMIC RAYS 119

Eq. (3.4) holds for both regions and that the most likely interaction for the photon is

scattering from a free electron39

(a) The solar core at R=R ¼ 0:025 (Table G.10).

(b) A diffuse region in the Galaxy of diameter, D ¼ 1 pc (T ¼ 105 K, ne ¼ 0:001 cm3).

3.9 (a) Write an expression for the fraction of hydrogen that is in the neutral state,

NHI=Ntot, for a gas in LTE. Evaluate all constants and express the result as a function of neand T.

(b) Consult Table G.11 to determine ne and T applicable to the Sun’s surface (i.e. letting

x ¼ 0 km) and, assuming LTE, evaluate NHI=Ntot.

(c) What can you conclude about the ionization state of hydrogen at the surface of the

Sun?

3.10 (a) Write an expression for the temperature, T as a function of energy level, n, for an

LTE situation in which the number of hydrogen atoms with electrons in level n is equivalent

to the number in the ground state.

(b) Using a spreadsheet or computer algebra software, compute T for the first 300 energy

levels of hydrogen. Plot TðnÞ.

(c) Briefly explain or comment on the behaviour of the plot.

3.11 (a) Find the factor by which the distance will be in error if dust extinction is ignored,

i.e. find f , such that dtrue ¼ derr f , where dtrue is the true distance to the object, and derr is the

distance calculated without taking AV into account.

(b) Suppose a F0Ib star is measured to have an apparent magnitude of V ¼ 4:5. IfAV ¼ 0:8, calculate the distance (pc) to the star, with and without the assumption of dust

extinction.

3.12 (a) Determine the ratio of the number of large classical (0:10 a 0:25 mm) dust

grains per unit volume in comparison to VSGs (0:005 a 0:01mm).

(b) If all of these grains have the same shape and mass density, determine the ratio of the

mass of classical grains in comparison to VSGs in some volume of space.

3.13 (a) Write Eq. (3.40) in the same manner as Eq. (3.41) but for the energy range,

102 E ðGeVÞ 4 106. That is, ensure that K is evaluated and use the appropriate

value of G. Assume that Eq. (3.41) applies at an energy of E ¼ 4 106 GeV.

(b) What is the total rate of CR particles (s1) in this energy range hitting the Moon? For

simplicity, assume that CRs are isotropic and all CRs hitting the Moon are absorbed.

39Note, however, that other types of interactions may be important depending on frequency (see Sect. 5.1.1.2).


Appendix: the electron/proton ratio in cosmic rays

We start by adopting a distribution in momentum, p, which is the same for both

electrons and protons, but with the possibility of a different constant of proportionality,

NCRðpÞ ¼ N0CR pG0 ð3:A:1Þ

where N0CR is the constant of proportionality and the subscript, CR, refers to the CR

species being considered (electrons, e, or protons, p). We also assume that the total

number of electrons over all kinetic energy,T , above a non-relativistic threshold,T0 ¼ 10

keV (Ref [147]), equals the total number of protons over the same energy range, i.e.Ntot.

For Ntot to be the same for both CR species, then we require an integration over the

distribution in energy, or the distribution in momentum, to equal this value,

Ntot ¼Z 1

T0

NCRðTÞdT ¼Z 1

p0CR

NCRðpÞdp ð3:A:2Þ

where p0CR is the lower momentum limit that corresponds to a fixed lower limit in

kinetic energy, T0, the latter being the same for both electrons and protons. Substituting

Eq. (3.A.1) for NCRðpÞ and integrating gives,

Ntot ¼N0CR

ðG0 1Þ p0G01CR

ð3:A:3Þ

Notice that, because the power law in momentum is a declining function, the limit

as p ! 1 goes to zero, so it is the lower momentum limit, p0CR, that determines

Ntot. Another way of stating this is that most of the particles are at low momentum

(low energy). Therefore, although we ultimately want to compare electron to proton

ratios at high energies, it is important for this integration to include the low energy

particles. Since both Ntot and G0 are the same for both electrons and protons, rearran-

ging Eq. (3.A.3) for the ratio of the electron to proton constants of proportionality

gives,

N0e

N0p

¼ p0ep0p

!G01

ð3:A:4Þ

We now require a relation between momentum and kinetic energy in order to relate

p0CR to a given fixed T0. The kinetic energy is just the total energy, E, less the rest mass

energy, so using Eq. (3.36),

T ¼ ðg 1ÞmCR c2 ð3:A:5Þ

APPENDIX: THE ELECTRON/PROTON RATIO IN COSMIC RAYS 121

where mCR is the rest mass of the CR particle. The momentum of a particle (see

Table I.1) is pCR ¼ gmCR v. Using Eq. ( 3.36) to eliminate the velocity, v, we find,

pCR ¼ mCR cffiffiffiffiffiffiffiffiffiffiffiffiffig2 1

pð3:A:6Þ

Using Eqs. (3.A.5) and (3.A.6) and eliminating g, we obtain a relation between kineticenergy and momentum,

pCR ¼ 1

c½T2 þ 2 T mCR c

21=2 ðall TÞ ð3:A:7Þ

E

cðT mCR c

2Þ ð3:A:8Þ

Eq. (3.A.8), for which Eq. (3.A.5) reduces to Eq. (3.36) (i.e. T E), is what is

expected for a relativistic particle40. The general expression, Eq. (3.A.7), indicates that,

at a fixed kinetic energy for both electrons and protons, the momentum of an electron

will be less than the momentum of a proton since me < mp. Therefore, p0e < p0p, so

the electron distribution is being sampled to a lower momentum than is the proton

distribution. Examination of Eq. (3.A.4) then requires that N0e < N0p. Using Eq.

(3.A.7) in Eq. (3.A.4) with the lower limit, p0CR, expressed in terms of T0, we obtain,

N0e

N0p

¼ T0 þ 2me c2

T0 þ 2mp c2

G01

2

ð3:A:9Þ

Since the lower kinetic energy limit was given as T0 ¼ 10 keV whereas the rest

energies, mCR c2, of an electron and a proton are much higher at 511 keV and

9:4 105 keV, respectively, this ratio reduces to,

N0e

N0p

¼ me

mp

G01

2

ð3:A:10Þ

Now we wish to compare the fraction of electrons to protons at high energies where

they are typically measured. At fixed high energy, E T , and we put Eq. (3.A.10) into

Eq. (3.A.1) using Eq. (3.A.8) to find,

NeðEÞNpðEÞ

me

mp

G01

2

ð3:A:11Þ

Using an initial spectral index of G0 ¼ 2:2, we findNeðEÞNpðEÞ 1 per cent which is

approximately what is observed (Ref. [147]).

40The momentum of a photon, for example, is expressed in this way (see Table I.1)


4Radiation Essentials

And God said, Let there be light: and there was light

– Genesis 1:3

4.1 Black body radiation

When a gas is in thermodynamic equilibrium (TE), the absorption and emission rates in

the gas are in balance. Such a situation could be set up if the interior walls of a box were

maintained at a single temperature, T. If no matter or radiation were permitted to escape

from the box, then the gas particles and all radiation within it would eventually reach an

equilibrium at this temperature, independent of the kind of material in the box or its

shape. It can be shown that the resulting radiation field will be isotropic and that its

spectrum (the emission as a function of frequency or wavelength) depends only upon T.

Such radiation is referred to as black body radiation (sometimes called ‘cavity

radiation’ because of the history of studying such radiation by setting up a ‘thermal

bath’ within a cavity). The resulting specific intensity is described by a particular

function called the Planck function or Planck curve (see the Appendix at the end of this

chapter for a derivation) which, in frequency-dependent and wavelength-dependent

forms, respectively, are given by,

BðTÞ ¼2h3

c2

1

ehkT 1

erg s1 cm2 Hz1 sr1 ð4:1Þ

BðTÞ ¼2hc2

5

1

ehckT 1

erg s1 cm2 cm1 sr1 ð4:2Þ


where cgs units have been specified. Plots of BðTÞ for different temperatures are

shown in Figure 4.1 and illustrate that, at higher temperatures, the specific intensity is

higher at all frequencies. The two functions are related to each other via,

B d ¼ B d ð4:3Þ

Thus, the integrals under the two curves, from zero to infinity, are equal. However, the

functions themselves are different. They have different maxima (Sect. 4.1.3) and

Eq. (1.5) must be used to convert from one form to the other. Since BðTÞ and

BðTÞ are specific intensities, the relations given in Chapter 1 involving specific

intensity also apply to the Planck function.

Since T in Eqs. (4.1) and (4.2) is a value that gives the specific intensity of the

radiation, it is a ‘radiation temperature’. However, we noted in Sect. 3.4.4 that for TE,

or LTE ‘locally’, the radiation temperature is equal to the kinetic temperature of the

gas. Thus, T in the above equations is written without subscript, indicating that it

represents kinetic temperature. Since there must be sufficient interactions between

radiation and matter for an equilibrium temperature to be established, this implies that

the gas must be opaque, as illustrated in Figure 4.2. For an object to be opaque, the

mean free path of a photon must be less than the size of the object. The observer sees

into the object over a distance about equal to the mean free path. We thus come to the

more formal definition of a black body, that is, a black body is an object that is a perfect

0

2e–05

4e–05

6e–05

8e–05

0.0001

0.00012

0.00014

0.00016

0.00018

0.0002

1e+13 1e+14 1e+15

)T(

un_

B

nu (Hz)

T=2000 K

T=4000 K

T=6000 K

T=8000 K

T=10000 K

Figure 4.1. Planck curves for temperatures from 2000 to 10 000 K. The ordinate axis is in units oferg s1 cm2 Hz1 sr1. Note that the peaks of the curves shift to higher frequency (smallerwavelength) as the temperature increases, and also that hotter objects have higher specificintensities than cooler objects at all frequencies.

124 CH4 RADIATION ESSENTIALS

absorber. If a photon, regardless of frequency, impinges on a black body, it will neither

be reflected back nor pass through unimpeded. The photon will be absorbed, implying

that its mean free path is less than the object’s size. Any opaque, non-reflecting object

at a single temperature is therefore a black body. This could include solids, provided

the solid material does not reflect light. For an object to remain at a single temperature,

absorptions and emissions must be in balance. Thus a black body must also be a perfect

emitter. Black bodies are therefore not black in colour nor does the word ‘black’ imply

that there is no emission. On the contrary, a black body emits the continuous Planck

spectrum whose shape is dictated by its temperature.

As we found for thermodynamic equilibrium (Sect. 3.4.4), identifying a perfect

black body in nature is difficult. However, there is one excellent example, that of the

cosmic microwave background (CMB) radiation that was shown in Figure 3.2. A

Planck spectrum is measured at every position on this map, the slight shifts in

temperature at different positions resulting in slightly different Planck curves at

different positions. The global mean Planck curve is given in Figure 4.3 and is well

fitted by a temperature of 2.73 K.

Stars are also very good examples of black bodies, although some departures from a

perfect curve do occur, as shown for the Sun in Figure 4.4. Even though the entire star is

not at a single temperature, the interior being hotter than the surface, we cannot see into

the interior (Figure 4.2) and, over the depth that can be viewed, the temperature is

approximately constant. Since stars have temperatures that correspond to Planck

curves that have peaks at or near the optical part of the spectrum, most of the

astrophysical radiation that we see with our eyes, whether naked eye or through optical

telescopes, is due to stars. This includes the Sun, of course, and it is unlikely to be an

accident that our eyes have the greatest sensitivity at just the region over which the

Solar Planck curve peaks (Figure 4.5). When we look out into the vast depths of space,

xObserver

x

l

Observer

Figure 4.2. In these figures, a single source is shining just behind a gas cloud of temperature, T,as seen by an observer on the right. In the top picture, the mean free path of a photon, l, is greaterthan the thickness, x, of the cloud. In such a case, the observer can see through the cloud to thesource. In the bottom picture, the mean free path of a photon is much smaller than the cloudthickness. The photon diffuses to the other side (similar to Figure 3.15), coming into equilibriumwith the temperature of the cloud. The observer will see into the cloud only as far as a mean freepath, i.e only into the outer ‘skin’ of the cloud, and will measure a Planck spectrum at thetemperature of this skin.

4.1 BLACK BODY RADIATION 125

Figure 4.3. Data points for this curve are measurements of the cosmic microwave backgroundradiation from a variety of sources that are listed on the plot. The solid curve is the best fit blackbody spectrum, corresponding to a temperature of 2.73 K (G.F. Smoot and D. Scott 2002, in TheReview of Particle Physics, K. Hagiwara et al., Phys. Rev. D66, 010001 (2002))

Figure 4.4. The Solar spectrum, shown as a function of increasing wavelength in a logarithmicplot. The plot is of flux density as measured from the distance of the Earth; to convert to specificintensity, a division by is required (see Eq. 1.13). The spectrum (black curve) is well fitted by ablack body at T ¼ 5781 K (grey shaded curve) but starts to depart significantly in the ultraviolet([0:3; mm due to numerous absorption lines. There are also significant departures in the X-rayand radio regions of the spectrum (not shown) due to Solar activity. These data are consistent witha Solar Constant (cf. Figure 1.6) of 1366:1Wm2.


it is starlight that we see, from our own Galaxy, as shown in Figure 3.3, to nearby

galaxies like those of Figure 3.8 and Figure 2.19, to distant galaxy clusters like the one

of Figure 7.6. Of all the emission mechanisms that we will discuss (see Part IV), it is

black body radiation, by far, that has the most relevance to our visual appreciation of

the sky.

If we can measure the specific intensity of an object and know it to be a black body,

then by Eqs. (4.1) and (4.2), we can determine its temperature. Recall that specific

intensity is independent of distance in the absence of intervening matter (see Sect. 1.3),

which means that we do not need to know anything else about the object, not even its

distance, to find T. Moreover, in principle, only a single measurement at one frequency

is required to give this result, although in practice, more measurements are usually

needed to ensure that the spectrum is indeed Planckian. The flip side of this coin is that,

since the Planck curve depends only on temperature, we can discern nothing about the

kind or amount of material that emits this radiation by studying its Planck curve alone.

Whether the black body is the glowing filament of an incandescent light bulb, ‘super

black’ man-made paint that reflects less than 1 per cent of the light that falls on it, or the

interior of a kitchen oven after an equilibrium temperature has been achieved, the same

Planck spectrum, dependent on T alone, will be measured.

4.1.1 The brightness temperature

Other than the CMB, most objects that we refer to as black bodies have emission

spectra that, like stars, are approximations to the Planck curve. To account for

departures from the Planck curve, we define the brightness temperature TB , at the

Figure 4.5. The Solar spectrum is again shown (middle curve, left scale) using the same data as inFigure 4.4 except over a more restricted wavelength range and on a linear scale. The brightnesstemperature (top curve, right scale) is also shown, derived by dividing the flux density by (Eq. 1.13) and then using the Planck formula (the wavelength form of Eq. 4.4) to recover TB. Thebottom curve, in arbitrary units, is the daylight photon flux response of the human eye smoothed to30 nm resolution (see also Figure 2.1). (Eye response reproduced by permission of James T. Fulton,2005, www.sightresearch.net/luminouseffic.htm)


observing frequency, , to be the temperature which, when put into the Planck formula,

results in the specific intensity that is actually measured at that frequency,

I ¼2h 3

c2

1

eh

kTB 1 ð4:4Þ

The wavelength form can also be used. It is clear that, if the same value of TB is

measured for all , then the object is a true black body. If TB is different for different

frequencies, then the object departs from a black body (Example 4.1). The extent to

which TB is not constant is a measure of how much the object departs from a black

body. The brightness temperature of the Sun as a function of wavelength is shown in

Figure 4.5. If the Sun were a perfect black body, then the brightness temperature curve

would be a straight horizontal line in this plot. This is close to being true over

the wavelength region shown, although there are some minor variations. However,

in the radio part of the Solar spectrum (not shown), the departures are much more

significant. For example, at 10 m, TB 5 105 K, almost two orders of magnitude

higher than shown in Figure 4.5, indicating that processes other than black body

emission are occurring (e.g. Sect. 8.5)1.

This definition of brightness temperature can be extended to include any astrophy-

sical source, whether or not its emission approximates that of a black body. For

example, TB could be specified even for an object whose spectrum is a power law

(e.g. Sect. 8.5). The brightness temperature simply provides a convenient way of

expressing the radiation temperature of an object, whether or not a physical (kinetic)

temperature is implied. Eq. (4.4) or its wavelength equivalent is a straightforward

functional relation between specific intensity and brightness temperature. Therefore, to

say that an object has a certain brightness temperature is essentially equivalent to

saying that it has a certain specific intensity, and vice versa.

Example 4.1

In a particular kind of telescope, the optical path is designed to switch back and forth betweenpointing at a target in the sky and pointing at a reference black body which is a rotating vaneused for calibration. An engineer must first fine tune the calibrator in the lab so that, over 2 cm to 20 cm, it emits as a true black body at a temperature of 50 K to within 2 K.After experimenting with different kinds of non-reflective paints and methods of obtaininga uniform temperature, he measures the specific intensity at the two end points ofthe wavelength range finding, I2 cm¼ 2:57 105erg s1cm2cm1sr1 and I20 cm ¼2:24 109 (in the same units). What is its brightness temperature at each wavelength?Is the calibrator a black body to within the desired tolerance?

1This emission refers to the radio quiet Sun. When there are radio bursts, the brightness temperature can be up to107 higher than this.


Since the units imply that the form of the Planck curve is being used, we rearrange

Eq. (4.2) to solve for the brightness temperature,

TB¼ h c

k

1

ln 1 þ 2hc2

5I

ð4:5Þ

From this equation, we find that the brightnesses temperatures at 2 cm and 20 cm are,

respectively, TB2 cm¼ 50:1 K and TB20 cm

¼ 43:4 K. The brightness temperature at 2 cm is

within the desired tolerance but the brightness temperature at 20 cm is not. Thus, the

calibrator is not yet a black body.

4.1.2 The Rayleigh–Jeans Law and Wien’s Law

TheRayleigh–Jeans Law andWien’s Law are not special ‘laws’ in a fundamental sense,

but are rather approximations to the Planck curve at the low and high frequency (or

wavelength) ranges. For example, suppose we observe the Planck curve at a low

frequency in comparison to the peak of the curve or, more technically, in the regime,

h kT or hkT

1. Then,

BðTÞ ¼2h3

c2

1

ehkT 1

2h3

c2

1

1 þ hkTþ . . . 1

¼ 22kT

c2ð4:6Þ

where we have used an exponential expansion (Eq. A.3). Note that higher orders are not

needed in the expansion because the term hkT

is less than 1 and is therefore diminish-

ingly small in higher orders. Eq. (4.6), or its wavelength equivalent (Prob. 4.2), is called

the Rayleigh–Jeans Law. The brightness temperature can also be expressed in this limit

by replacing BðTÞ with I and T with TB in Eq. (4.6).

On the other end of the spectrum, where hkT

1, the Planck function can be

written,

BðTÞ ¼2h3

c2

1

ehkT 1

2h3

c2ðeh

kTÞ ð4:7Þ

In this regime, the exponential term dominates over 3 so the function diminishes

rapidly as the frequency increases. Eq. (4.7), or its wavelength equivalent (Prob. 4.2), is

called Wien’s Law.

Examples of the two approximations are shown in Figure 4.6. Example 4.2 indicates

when they are valid to within 10 per cent.


Example 4.2

For what values of hkT do the Rayleigh–Jeans and Wien’s approximations depart from the

Planck curve by more than 10 per cent?

This can be determined by letting x ¼ hkT

and setting the ratio of Eq. (4.6) over Eq. (4.1)

equal to 1.1, i.e. ðex 1Þ=x ¼ 1:1. The solution to this equation is x ¼ 0:19. For Wien’s Law,

we set the ratio of Eq. (4.7) over Eq. (4.1) to 0.9 since Wien’s approximation gives values

lower than the full Planck equation, exðex 1Þ ¼ ð1 exÞ ¼ 0:9. In this case, x ¼ 2:3.

Therefore, the use of the Rayleigh–Jeans law will give an accuracy of better than 10 per cent,

provided hkT

< 0:19 and Wien’s law gives better than 10 per cent accuracy if hkT

> 2:3.

4.1.3 Wien’s Displacement Law and stellar colours

To find the wavelength, or frequency, of the peak of the Planck curve, we need only

differentiate Eq. (4.1), or Eq. (4.2), respectively, and set the result to zero, the result

being (cgs),

maxT ¼ 0:290

max ¼ 5:88 1010 Tð4:8Þ

1e–06

1e–05

1e–4

1e14 1e15

B_n

u(T

)

nu (Hz)

Wien

’s

Ray

leig

h-Je

ans

Figure 4.6. A Planck curve, in cgs units, for T ¼ 10 000 K (solid curve) showing the Rayleigh–Jeans and Wien’s approximations (both dashed).


This is called Wien’s Displacement Law2. It indicates how the peak of the Planck curve

shifts with wavelength or frequency. Note that max 6¼ c=max since the wavelength and

frequency forms of the Planck curve are different functions, as noted in Sect. 4.1.

Figure 4.1 illustrates that, as the temperature increases, not only does the specific

intensity increase, but also the peak of the curve shifts to higher frequencies as Wien’s

Displacement Law predicts.

These simple relationships provide very powerful tools for obtaining the temperature

of a black body. While only a single measurement of BðTÞ, in principle, is required to

determine T (Sect. 4.1), for many objects, stars in particular, the specific intensity

cannot be measured because the object is unresolved (see Figure 1.9.). In such cases, it

is the flux density that is measured instead. However, the flux density of the black body,

as observed by a telescope, is related to its specific intensity via f ¼ BðTÞ, where is the solid angle subtended by the object (Eq. 1.13). Although is not known for an

unresolved object, it is assumed not to vary with frequency3, which means that the

shape of BðTÞ will be the same as the shape of f . We show an example in Figure 4.5 in

which it is the flux density of the Sun, rather than its specific intensity, that is plotted.

For the Sun, we know , but we could easily plot f for any star for which the solid

angle is not known. By simply identifying the frequency of the peak of the spectrum

and using Wien’s Displacement Law, the temperature can be found. Note that this can

be done without knowing the distance to the star.

Since most of the emission of any black body occurs around the peak of the curve,

Wien’s Displacement Law also explains how the colour of a hot opaque object changes

with its temperature. Subtle colour differences of stars are quite visible to the naked eye.

Examples can be found throughout the sky, but a good contrast is between the reddish

star, Betelgeuse in Orion’s shoulder and the blueish star Rigel in his foot (see Figure 5.8

for locations). The shift in the peak of the Planck curve provides the basis for the colour

index used in the observational HR diagram shown in Figure 1.14 and discussed in

Sect. 1.6.3. The ratio of Planck curve values at two different frequencies, or equivalently

the ratio of two flux densities, gives a colour index via Eq. (1.32). The colour index of a

star of unknown angular size and distance can be compared to (calibrated against) the

colour index based on the ratios of Planck curves of known temperature in order to relate

colour index to stellar temperature (see Example 4.3). Such a calibration is shown in

Figure G.1 of Appendix G. Once a good calibration has been set up, one need only make

two measurements of stellar flux density in order to determine the surface temperature.

This procedure is even simpler than determining the peak of the spectrum, since to find

the peak, a number of measurements would be required. A caveat is that stellar spectra

do show departures from a perfect black body as Figure 4.4 illustrates. Thus, tempera-

tures determined by different techniques are not always exactly the same. Also, the

calibrations given in Tables G.7, G.8 and G.9 do not include the finer subclasses that are

present over a continuum of stellar temperatures.

2This should not be confused with ‘Wien’s Law’ of Sect. 4.1.2.3Small changes in can occur with frequency, but these are generally not sufficient to make significant changes inthe shape of the spectrum.


Example 4.3

The star, Gomeisa (CMi) is measured to have B ¼ 2:79 and V ¼ 2:89. Determine its colourindex, B V, the ratio of its specific intensities at the two observing wavelengths,BBðTÞ=BVðTÞ, estimate its surface temperature, T, determine the wavelength of the peak ofits Planck curve, max, and provide a best estimate of its spectral type, given that it is a mainsequence star. Assume that there is no dust extinction in the line of sight.

From the measurements, B V ¼ 2:79 2:89 ¼ 0:10. Using Eq. (1.32) for colour

index, 0:10 ¼ 2:5 logfBfV

ð0:601 0Þ, where we have taken the zero point value

from Table 1.1 and are using the wavelength form of the Planck curve since we are being

asked for max. Solving for the ratio of flux densities,fBfV¼ 1:9. This result is the same as the

ratio of specific intensities, BBðTÞBVðTÞ, because the solid angle subtended by the source is

considered constant with wavelength. Referring to the calibration of Table G.7 for main

sequence stars, B V ¼ 0:1 corresponds to T ¼ 11950 K and a spectral type of B8. By

Wien’s Displacement Law (Eq. 4.8), the spectrum would peak at max ¼ 243 nm. As a

check on the ratio of specific intensities, we can use the above temperature in the Planck

equation (Eq. 4.2) and re-compute the ratio. Using the central wavelengths for B and V

bands (Table 1.1), we find, BBð11950ÞBVð11950Þ ¼ 1:7 which is within 10 per cent of the value of 1.9

above and within 5 per cent of it when allowing for a range of temperatures between

spectral types B5 and A0 in the table.

4.1.4 The Stefan–Boltzmann Law, stellar luminosity and the HR diagram

The total intensity of a black body can be found by integrating under the Planck curve

over all frequencies, or all wavelengths,

BðTÞ ¼Z 1

0

BðTÞ d ¼Z 1

0

BðTÞ d ¼ KT4 erg s1 cm2 sr1 ð4:9Þ

where,

K ¼ 2ðkÞ4

15c2h3ð4:10Þ

If we now use the relationship between intensity and flux given in Eq. (1.14) for

emission escaping from a surface over 2 sr, then the resulting astrophysical flux

through the surface of such a black body is,

FBB ¼ T4 ð4:11Þ


where ¼ K ¼ 5:67 105erg s1 cm2 K4 is the Stefan–Boltzmann constant.

Eq. (4.11) is usually written in the more familiar form,

F ¼ Teff4 erg s1 cm2 ð4:12Þ

This equation defines the effective temperature, Teff , which is whatever temperature

results in the flux observed from the object. Like the brightness temperature, the

effective temperature allows for departures from a perfect black body. In the former

case, there could be a different brightness temperature at each wavelength, but in the

latter case, there is only one effective temperature for any object. Also, although a

brightness temperature may be quoted for any object, whether approximating a black

body or not, the effective temperature is usually only quoted for objects whose

emission resembles a Planck curve, especially stars. Eqs. (4.11) or (4.12) are referred

to as the Stefan–Boltzmann Law.

Using this flux, it is straightforward to calculate the total (i.e. bolometric) luminosity

of the object which, for a sphere of radius, R, is (see Eq. 1.10),

L ¼ 4R2T4eff ð4:13Þ

It is now clear that the luminosity of a star or any black body is a strong function of its

temperature. This is why, for any mixture of stars, the luminosity tends to be dominated

by hot, high-mass stars (Prob. 4.7).

This equation also helps to clarify the placement of the stars on the HR diagram, an

example of which is shown in Figure 1.14. Taking the logarithm of both sides of

Eq. 4.13, logðLÞ ¼ 4 logðTeffÞ þ 2 logðRÞ þ constant. The log of the luminosity is

related to the absolute visual magnitude (see Eqs. 1.31 and 1.33) which is the ordinate

of the HR diagram. The temperature, Teff is related to the colour index, B V, as

discussed in the previous section. The HR diagram, therefore, can be understood as a

relationship between the luminosity and temperature of the star.

If all stars were the same size, then a plot of logðLÞ against logðTeffÞ would yield a

straight line of slope, 4. Although Figure 1.14 does not show exactly these axes, it is

clear that stars are not randomly distributed in the HR diagram. For the main sequence,

we see that cooler stars on the right are also much dimmer than the brilliant hotter stars

on the left as Eq. (4.13) predicts. In fact, a plot of logðLÞ versus logðTeffÞ yields a main

sequence slope that is steeper than 4 indicating that, along the main sequence, hotter

stars are also larger. This is shown more quantitatively in Table G.7. Some second

order effects are also present4, causing curvature in the main sequence.

There are many stars in the upper right region of the HR diagram as well.

For the same temperature, these stars are much more luminous than their main

sequence counterparts. By Eq. (4.13), this means that they must be much larger,

hence the designation, ‘giants’. A G2III star, for example (Table G.8) has about the

same surface temperature as the Sun, yet is ten times larger. The internal structure of

4Stellar opacity (see Sect. 5.4.1) can affect the radius of the star, for example.


such stars is different from that of those on the main sequence, as discussed in

Sect. 3.3.2. Notice, however, that some very strong conclusions can be drawn about

stars by simply plotting their location on the HR diagram and referring to Eq. (4.13).

It is from these and other observational starting points that we can actually piece

together the evolution of stars.

4.1.5 Energy density and pressure in stars

The approximation that stars are close to black bodies also implies that the radiation

field is close to (but not exactly) isotropic at any interior location (see Sect. 6.3.6 for a

more accurate description of radiation within stars). Letting the intensity, I, be

approximately equivalent to the mean intensity, J, for such a case, we can use Eq.

(1.16) to write the energy density (erg cm3) as,

u ¼ 4

cI ¼ 4

cB ¼ 4

cKT 4 ¼ aT 4 ð4:14Þ

where we have used Eq. (4.9) and the constant, a ¼ 7:57 1015erg cm3 K4, is

called the radiation constant whose accurate value is given in Table G. 2. Thus, the

energy density is also a strong function of temperature.

The radiation pressure (dyn cm2) of such a gas is then given by (Eq. 1.22),

Prad ¼ 1

3aT 4 ð4:15Þ

This radiation pressure must be added to the particle pressure (Eq. 3.11), as described

in Eq. (3.16), to determine the total pressure at any location within a star. For low mass

stars, like our Sun, the radiation pressure is negligible with respect to the total.

However, for hotter more massive stars, the radiation pressure, being a strong function

of temperature, becomes dominant. For very high mass stars, for example, radiation

pressure can become so important that it affects the stability of stars and contributes to

mass loss of the outer layers (e.g. see the discussion of Wolf–Rayet stars in

Sect. 5.1.2.2). The importance of radiation pressure for stellar dynamics will be

discussed further in Sect. 5.4.2.

4.2 Grey bodies and planetary temperatures

Opaque bodies at a single surface temperature are black bodies if they absorb all

incident radiation. However, in many cases, some fraction of the incident radiation will

be reflected away. The fraction of light that is reflected is called the albedo, A,5 of the

5From the latin, ‘albus’ for white.


object, values of which are given in Table G.4 for the planets. A surface that absorbs all

incident radiation (a true black body) would have an albedo of 0 and a surface that

reflects all incident radiation (such as a ‘perfect’ mirror) has an albedo of 1. We

implicitly considered ‘perfect absorption’ and ‘perfect reflection’ previously when

dealing with radiation pressure in Sect. 1.5. The introduction of the albedo now permits

a quantification of intermediate cases. Objects that do not absorb all incident radiation

are called grey bodies. The term, ‘grey’, is meant to imply that the absorption is

independent of wavelength (i.e. ‘grey’ or ‘colourless’). This is technically only true for

opaque objects that reflect the same fraction of light at any wavelength, i.e. A is

independent of , but the terminology also tends to be used for any object with A>0

(see Figure 4.7)6.

Planets, asteroids, comets, moons (see Iapetus, Figure 4.8), and dust grains (see also

Appendix D.3) are all grey bodies, as are normal opaque objects in every day life like

the walls of a room, the surface of a human being, or the hound dog shown in

Figure 4.9. It is the reflected light that allows us to see the object. Our eyes detect

the portion of the reflected light from walls, dogs, or planets that falls into the visual

band. It is the absorbed light that heats the object. The object will then radiate with a

Planck spectrum dictated by its temperature.

Figure 4.7. Plot of the Earth’s albedo as a function of wavelength in the optical band, measuredon 19 November 1993 at the times indicated. The albedo increases towards the blue (left) due toRayleigh scattering (see Sect. 5.1.1.3) and a number of spectral features can be seen, including oneat 760 nm due to O2. These wavelength-dependent values are comparable to the Bond Albedo ofA ¼ 0:306 (Table G.4) which is the value applicable to all wavelengths. (Montanes-Rodriguez, P.,et al., 2005, ApJ, 629, 1175. Reproduced by permission of the AAS)

6This is because, for many such objects, the albedo does not change drastically over the optical part of thespectrum, as Figure 4.7 shows. The albedo in the optical band is most important because this is where the Sun’sPlanck curve peaks and therefore is the band over which most of the energy is absorbed.

4.2 GREY BODIES AND PLANETARY TEMPERATURES 135

4.2.1 The equilibrium temperature of a grey body

Grey bodies will absorb a fraction, 1 A, of the light that falls on them and it is this

portion that heats the object, bringing it to a temperature that remains constant with

time, that is, an equilibrium temperature. For such an equilibrium temperature to be

Figure 4.8. Image of Iapetus, a moon of Saturn (equatorial diameter¼ 1436 km), taken at ‘half-moon’ phase from the Cassini spacecraft in October, 2004, with a resolution of 7 km. Iapetus isunique in the Solar System because of the strong contrast in albedo between its two hemispheres.This constrast can be seen here as dark and bright sections on the illuminated side of the moon.(The narrow white streak in the dark region is a chain of icy mountains.) The dark section is as darkas coal (A ¼ 0:03 ! 0:05, almost a black body) while the bright section is more typical of Saturn’sother icy moons (A ¼ 0:5 ! 0:6). The reason for this contrast is not yet understood. (Reproducedcourtesy of NASA/JPL/Space Science Institute)

Figure 4.9. This hound dog is a ‘grey body’. A fraction of the light that falls on the dog isabsorbed and the remaining fraction is reflected. It is the reflected light in the optical part of thespectrum that allows us to see the dog normally as shown on the left. The absorbed light as well asthe dog’s internal heat generation must be balanced by its emission to maintain a constanttemperature at any given position. The emitted spectrum will be a Planck curve which peaks in theIR for the range of surface temperatures shown in the false colour image on the right. (Reproducedby permission of SIRTF/IPAC) (See colour plate)


maintained, the rate at which the grey body radiates must balance the rate at which it

absorbs energy. Since a fraction of the incident energy is reflected away, the tempera-

ture of a grey body will not come into equilibrium with the temperature of the radiation

field but rather with a temperature that corresponds to the absorbed fraction of the

incident radiation. The grey body then emits a Planck spectrum at its equilibrium

temperature. Example 4.4 shows how the equilibrium temperature of an externally

heated grey body can be determined. Emissions from grey bodies in our Solar System

have Planck curves that peak in the infrared (Prob. 4.6). Dust that is detected in our

Milky Way, such as in the Lagoon Nebula shown in Figure 3.13, as well as the dust that

is observed in other galaxies such as in Arp 220 (Figure 4.10) also tend to be at

temperatures resulting in max in the IR or sub-mm.

There are two cases, however, in which a grey body may have a temperature that

departs from the equilibrium temperature expected from absorbed external radiation.

The first is if the object has its own internal source of heat. For living, warm-blooded

creatures, for example, chemical reactions turn food into an internal heat source.

Figure 4.10. Image of the ultra-luminous infrared galaxy, Arp 220. This galaxy is undergoing anintense ‘starburst’, meaning that it is rapidly forming stars, in this case, at a rate of 100Myr

1.Thus, there are many hot young stars in this galaxy (see Sect. 3.3.4) whose UV photons heat theabundant dust that permeates the system. (An active nucleus is also present.) Most of the galaxy’sluminosity is then re-radiated in the IR by dust grains, giving the very high value ofL8!1000 mm ¼ 1:6 1012 L. Arp 220 is believed to be the remnant of two galaxies that havemerged. The inset shows a Planck spectrum that has been fitted to the dust emission from thegalaxy. The spectrum is adjusted slightly from a true Planck spectrum to take into account grainproperties. [Reproduced by permission of David Sanders. (Adapted from Lisenfeld, U., Isaak, K.G.,and Hills, R., 2000, MNRAS, 312, 433)]


For the outer gas giant planets, Jupiter, Saturn, and Neptune, there are significant

internal heat sources7, possibly remnant heat from their formation (Jupiter) or heat

released by the precipitation and slow sinking of helium droplets towards the core

(Saturn). Internal heat can also be generated via tidal stresses, such as those that occur

within Jupiter’s close moon, Io, due to the gravitational forces exerted on it by Jupiter.

For the Earth, an internal heat source is the radioactive decay of elements in the crust.

However, this source, with a flux of Sint ¼ 0:06 W m2 (Ref. [174]), is insignificant in

comparison to the incident Solar flux, i.e. that of the Solar constant, S ¼ 1367 W m2

(see Sect. 1.2, Figure 1.6).

The second is if the heat from the planet that would normally be radiated away freely

is somehow trapped. This is what happens from the Greenhouse Effect. Planets such as

the Earth and Venus are shrouded in an atmosphere that acts like a blanket, retaining

heat within it. This occurs because most incident Solar radiation is at optical wave-

lengths (see Figure 4.5) whereas the emitted radiation from the surface is in the

infrared. The atmosphere is not as transparent in the infrared as in the optical, as the

atmospheric transparency curve for the Earth (Figure 2.2) shows. A photon emitted

from the surface may be scattered in the atmosphere or re-absorbed and re-emitted,

possibly to be absorbed by the surface again. Thus, the atmosphere has the effect of

retaining photons for a longer period of time, contributing an extra source of heating.

The Greenhouse Effect provides an important and necessary heat source to the Earth,

contributing þ33C to its surface temperature, as Example 4.4 notes. ‘Global warm-

ing’ refers to the much smaller changes, of order less than a degree Celsius, that the

Earth has been experiencing over the last century, as Figure 4.12 illustrates.

Example 4.4

Estimate the equilibrium temperature of the Earth. Compare this to the Earth’s globallyaveraged mean temperature of þ14C (287 K).

The Earth absorbs radiation over only the cross-sectional area, R2, that faces the Sun

(see Figure 4.11 and Example 1.1). The rate at which energy is absorbed is therefore

Labs ¼ ð1 AÞR2f , where f is the flux of the Sun at the distance of the Earth (i.e. the

Solar Constant) and A is the Earth’s albedo. Expressing the Solar flux in terms of

luminosity, L ¼ 4r2f (Eq. 1.9), where r is the Earth–Sun distance, the above equation

becomes,

Labs ¼ð1 AÞR

2L

4r2ð4:16Þ

The Earth can be approximated as a ‘fast rotator’, which means that the energy

absorbed on the daytime side is quickly equalized over the nighttime surface. This is

7These outer planets radiate significantly more energy (of order a factor of 2) than they are absorbing from theSun. By contrast, the temperature of the planet, Uranus can be explained solely by external Solar heating.


a reasonable approximation because day/night variations in surface temperature (of

order 10 K) are much smaller than the temperature itself (287 K). The Earth, being a

grey body, then emits a Planck spectrum at a (globally averaged) temperature, T. The

rate of energy emission is given by the Stefan–Boltzmann law (Eq. 4.13)8,

Lem ¼ 4R2T4 ð4:17Þ

For an equilibrium temperature, we require the absorption rate to equal the emission

rate, so equating Eq. (4.16) to Eq. (4.17) and solving for temperature,

T ¼ ð1 AÞL16r2

1=4

ð4:18Þ

Figure 4.11. A planet receiving light from the Sun intercepts it over its daylight half (left). Beinga grey body, a fraction, A (the albedo), of this light is reflected away and a fraction, 1 A, isabsorbed. Assuming that the absorbed energy is equalized quickly over the planet’s surface, theplanet then radiates the energy over its entire surface (right) at a rate that balances the energyinput rate, thus maintaining an equilibrium temperature (see Example 4.4).

Figure 4.12. Globally averaged temperature anomaly of the Earth from 1880 to 2004, with respectto a zero point of 14C that is an average over the period, 1951 to 1980 (Reproduced by permissionof J. Hansen, from http://data.giss.nasa.gov/gistemp/). The increase in temperature correlateswith an increase in carbon dioxide concentration in the atmosphere.

8A multiplicative factor is actually required on the right hand side of this equation to account for the emissiveproperties of the surface material, but the correction is small for the purposes of this example: approximately 0.9for solid surfaces and 0.65 for gaseous surfaces (Ref. [75]), leading to corrections to T (Eqn. 4.18) of 0.97 and0.90, respectively.


Using A ¼ 0:306 (Table G.4), L ¼ 3:845 1033erg s1; r ¼ 1:496 1013 cm

(Table G.3), and ¼ 5:67 105erg cm2 K4 (Table G.2), we find T ¼ 254K ¼19C. By Wien’s Displacement Law (Eq. 4.8), the peak of the corresponding Planck

curve would be in the infrared (max ¼ 11mm). Note that Eq. (4.18) is independent of

the size of the Earth,R. Thus, provided the above assumptions are valid, a planet of any

size placed at the distance of the Earth would have the same equilibrium temperature if

it has the same albedo. This equilibrium temperature is considerably colder than the

Earth’s measured globally averaged temperature of T ¼ 287K ¼ þ14C (Ref. [29]),

the increase being due to the Greenhouse Effect. It is clear that, without the warming

blanket of our atmosphere, the Earth would be a very cold, inhospitable place to live.

Eq. (4.18) in Example 4.4 can be used for any grey body in the Solar system whose

illumination and radiation geometry are as shown in Figure 4.11. Changes in the

geometry of a grey body will not make large changes in the results, however. For

example, a hypothetical asteroid shaped like a cube, rather than a sphere, would have

an equilibrium temperature that differs from that given by Eq. (4.18) by a factor of

only 1.1. A ‘slow rotator’ – an object that cannot equalize its temperature between its

day/night sides, but rather is hot on one side and cold on the other – will have a hot side

temperature that is higher than that of Eq. (4.18) by a factor of 1.2. Therefore, Eq. 4.18

provides a good estimate of the temperature of a grey body that is heated only by

the Sun.

If the appropriate external luminosity is used instead of L, Eq. (4.18) can also be

used for a grey body in an external system such as dust in the ISM or other galaxies9,

dust rings around stars such as that shown in Figure 4.13 or, as discussed below, planets

orbiting stars other than our Sun (extrasolar planets).

4.2.2 Direct detection of extrasolar planets

Almost all extrasolar planets (or exoplanets) have been discovered by observing the

perturbation in the motion of the parent star due to the orbital motion of the planet

around it (see Sect. 7.2.1.2). Thus, it is the stellar emission that is actually observed and

the planet is inferred to be present by the dynamical effect it has on the parent star. It is

now also possible to infer the presence of exoplanets in spatially unresolved systems

via gravitational microlensing, a topic that will be further explored in Sect. 7.3.2.

However, microlensing is a relatively rare event that requires long term monitoring of

many stars. It is of particular interest, then, to see if the ‘normal’ Planck emission from

a planet can be detected directly.

9Some modification is required, however, if the grey body is surrounded on many sides by a number of stars, suchas would be the case for dust grains in the interstellar medium. Also, the smallest dust grains do not achieve anequilibrium temperature, but rather experience spikes in temperature due to the absorption of individual photons,followed by rapid cooling.


There are two challenges to overcome. The first is that an entire planetary system

is often unresolved, so the flux density of the planet(s) and star are received

together in a single beam or pixel. For example, if a relatively close star at a

distance 5 pc had a planet orbiting it at the same distance as Jupiter is to the Sun,

then the planet and the star would be separated by only 1 arcsecond. The second

challenge is that the star is much hotter (and also larger) than the planet, so the

starlight swamps the emission from the planet itself. The total flux density received

at any wavelength, for an unresolved system, is the sum of the flux densities from

the planet and star together, but the perturbation (increase) in flux due to the planet

is very small, even at the infrared wavelengths at which the planet shines brightest,

as Example 4.5 shows. Even if a perturbation to the star’s spectrum could be

measured, it would not necessarily be clear that this perturbation is due to a planet.

Perhaps the star has a disk of dust around it, for example, or perhaps the star’s

spectrum is unusual in some way. Fortunately, there are two classes of extrasolar

planet that do lend themselves to direct detection: those for which the planet is

very close to the parent star, and those for which the planet is very far away. In

both cases, the planet must be fairly massive, of order several Jupiter masses rather

than the mass of the Earth.

Such extrasolar planets that are close to the parent star are called hot Jupiters. These

planets are as massive as Jupiter, but orbit their stars at a very small radius, of order

0.05 AU, and have revolution periods of only a few days. Due to their proximity to the

star, these planets will be very hot and bright. In addition, their closeness to the star

Figure 4.13. A dust disk observed around the star, HR 4796A, by the Hubble Space Telescopeusing its NICMOS instrument in the near infrared. The brilliant starlight has been blocked out bycovering over a region shown by the circle in this picture, revealing the faint disk around it. Thecentral star is located at the centre and dashed curves are interpolations as to where the diskcontinues through the blocked region. Neptune’s orbit is drawn at the bottom right for comparison.(Reproduced by permission of G. Schneider)


makes it much more likely that they will pass directly in front of or behind the star, in

their orbit around it, as seen from our vantage point. The resulting eclipse can make a

measurable difference in the light from the system. In such a case, it is clear that the

slight change in flux is due to the planet passing in front of or behind the star and not to

a dust disk which would always be visible. The first two detections of hot Jupiters in

eclipse were made in the year, 2005. The light curve (a plot of varying emission with

time) of one of these is shown in Figure 4.14.

The second class involves direct imaging. For planets that are very far from their

parent star, it is now becoming possible to image the planet directly. The planet appears

only as a weak ‘pin-prick’ of light that is sufficiently separate from the stellar image to

be seen as distinct. Such imaging is now becoming possible as detectors improve and

observing techniques, such as those that deal effectively with speckles (see Sect. 2.3.2),

are put into practice. Two early observations made in 2004 and 2005 are of planetary

candidates that are each a few Jupiter masses and at distances of 55 AU and 100 AU

from their parent stars, respectively (Ref. [38], Ref. [110]). As telescopes and detectors

continue to improve, such detections will soon become routine.

Figure 4.14. Light curve from the extrasolar planet system, TrES-1. At the centre of the plot, theflux is lower because the planet is behind the star. At the right and left, the flux includes boththe planet and star. These observations were taken at 8mm with NASA’s Spitzer orbiting telescope(Ref. [36]). They constitute one of the first two observations, published in 2005, in which photonsfrom a planet have been directly and clearly detected (see also Ref. [45]). The planet in this systemis one of a class of planets called hot ‘Jupiters’. The star is of type, K0V and the planet is a distance,r ¼ 0:0394 AU, from it with an albedo of A ¼ 0:31 and mass, Mp ¼ 0:76MJ, where MJ is the mass ofJupiter. (Reproduced by permission of D. Charbonneau)


Example 4.5

A Sun-like star at a distance of d ¼ 5 pc has an Earth-like planet orbiting 1 AU from it.Determine the excess flux density due to the planet at the wavelength of the peak of theplanet’s Planck curve. Assume that the planet has no atmosphere and that the system isunresolved.

A Sun-like star has a temperature of 5781 K (Table G.3) and, from Eq. (4.18), we know

that an Earth-like planet without an atmosphere (no Greenhouse Effect) will have a

temperature of 254 K (Example 4.4). By Wien’s Displacement Law for the planet

(Eq. 4.8) max ¼ 1:14 103 cm. Using Eq. (4.2) at this wavelength, the star’s specific

intensity is, Bsð5781Þ ¼ 2:53 1010erg s1 cm2 cm1sr1 and the planet’s specific inten-

sity is Bpð254Þ ¼ 4:32 107erg s1 cm2 cm1 sr1. The flux density of each is

f ¼ BðTÞ (Eq. 1.13). The solid angle, ¼ ðR=dÞ2(Eq. B.3 and B.1), where R is

the radius of the object. Using the radius of the Sun (Table G.3) and the Earth (Table G.4),

the star’s solid angle is s ¼ 6:39 1017 sr and the planet’s is p ¼ 5:37 1021 sr.

Then the flux density of the star and planet are, respectively, fs ¼ 1:62106erg s1 cm2 cm1 and fp ¼ 2:32 1013erg s1 cm2 cm1. The total flux density

detected is fs ¼ fs þ fp fs. The planet would therefore make a very tiny perturbation that

is only 1:4 107 times the value of the star’s flux density at a wavelength of max for the

planet. In order to detect such a tiny perturbation, the dynamic range (see Sect. 2.5) of the

observations would have to be better than 7 106.

Problems

4.1 (a) Verify that Eq. (4.1) can be reproduced when Eq. (4.2) is substituted into Eq. (4.3).

(i.e. show that BðTÞ ¼ BðTÞðd=dÞ.

(b)Using Wien’s Displacement Law, evaluate max and max for a temperature of 6000 K

and show that max 6¼ c=max.

(c) Evaluate Bð6000Þ at ¼ max.

(d) Evaluate Bð6000Þ at the frequency corresponding to max. Convert the result to the

same units as Bð6000Þ and verify that the result is the same as in part (c).

4.2 (a) Write the Rayleigh–Jeans Law and Wien’s Law in their wavelength-dependent

forms. Will the criteria specified in Example 4.2 for departures from the full Planck curve

be the same for the wavelength-dependent functions as for the frequency-dependent

functions?

(b) Assuming that the emission can be well-approximated by a Planck curve, determine

whether the Rayleigh–Jeans Law or Wien’s Law may be used for the following conditions:


(i) an observation of the Sun at 912 A;

(ii) an observation of a dense molecular cloud of temperature, T ¼ 20 K at ¼230 GHz;

(iii) an observation of the 2.7 K CMB at 21 cm.

4.3 (a) Determine and plot on the same graph, BðTÞ for the eye region (closest to the

camera) and the nose of the hound dog shown in Figure 4.9, using the scale in the figure to

estimate the temperature.

(b) Determine max (mm) for the same eye and nose.

(c) Assuming that photons leaving both the eye and nose can escape from the surface in

all directions outwards (over 2 sr), determine the flux density at the surface of the eye and

nose.

(d) Is the energy loss rate greater through the eye or through the nose of the hound dog?

4.4 Repeat Prob. 4.3 but, instead of the eye and ear of a hound dog, compare a O8 V star

to a K7 V star (Table G.7). Indicate in which part of the spectrum max occurs for each.

4.5 In the caption to Figure 4.1, we note that a hot object is brighter than a cool object at all

frequencies when both emit a Planck spectrum. Yet, observations of the ISM in the infrared

show mostly cool dust emission while the hotter stellar emission is negligible in comparison.

How can these two statements be reconciled? To answer this question, consider a cubic region

of volume, 1pc3 in the Galaxy some distance,D, away so that any object in the volume can be

considered at the same distance. Assume that there is, at most, one star within this volume and

that this star is equivalent to the Sun (same radius, R, and effective temperatue, Teff). Assume

that all dust grains are spherical, evenly distributed, and that no grain ‘shadows’ (is directly in

front of) any other grain. For the dust, let the temperature, T ¼ 100 K, radius, a ¼ 0:2m,

density, ¼ 2:5 g cm3, and the total dust mass in the volume is 2:65 104 M. Find the

ratio of the stellar flux to the dust flux at IR (100 mm) and visible (550 nm) wavelengths,

assuming both emit perfect planck spectra, and comment on the results.

4.6 Compute the temperature range over which an object would have a Planck curve

peak, max, in the infrared.

4.7 Assume, for simplicity, that a region of space contains one O8V star, one A0Ib star,

100 G5V stars and 1000 M2V stars. Refer to Tables G.7, G.8, G.9 and,

(a) determine the total stellar mass (units of M) in the region and indicate which type of

star dominates the mass;

(b) determine the total stellar luminosity (L) in the region and indicate which type of

star dominates the luminosity.

4.8 (a) Write an expression for the ratio of the flux of a star to the flux of the Sun, f?=f, in

terms of the radii of these objects and their effective temperatures. Assume that the two are

at the same distance and radiate as perfect black bodies.


(b) Using the results of (a), find the flux ratio for the neutron star, J0538þ2817, whose

radius is R ¼ 1:68 km and temperature is T ¼ 2:12 106 K. Determine max for this

neutron star and indicate in what part of the spectrum this occurs.

(c) Presently, thermal black body radiation has been measured from the surface of only a

‘handful’ of neutron stars (Ref. [84])10. Based on the results of part (b), explain why this

might be the case.

4.9 Compute the equilibrium temperature of Mars. Compare it to the average measured

temperature of 63C and comment on whether Mars has its own internal heat source or

experiences a Greenhouse Effect.

4.10 (a) Rewrite Eq. (4.18) for a grey body that is a slow rotator, i.e. determine T for the

Sun-facing side of a spherical body when the temperature is negligible on the hemisphere

opposite to the Sun.

(b) On Iapetus (see Figure 4.8), the bright and dark sections are often separated by no more

than a single pixel of width, 7 km. If an astronaut were to take a stroll over such a distance at

the bright/dark boundary, by how much would the surface temperature change? Which is the

warm side? Assume that Iapetus is a slow rotator and its heat is supplied only by the Sun.

4.11 From the information given in Figure 4.14 for the extrasolar planet system, TrES-1,

as well as Table G.7, find the following:

(a) the temperature, Ts, radius, Rs, and luminosity, Ls, of the star ;

(b) the temperature, Tp, of the planet;

(c) the ratio of flux densities of the planet to the star, fp=fs at the observing wavelength

of 8 mm;

(d) the ratio of specific intensities, BðTpÞ=BðTsÞ at 8 mm;

(e) the ratio of solid angles, p=s;

(f) the radius of the planet, Rp. Express this result in terms of Jupiter radii, RJ.

4.12 (a) Derive an expression for the reflected flux, Frefl, of an extrasolar planet that

intercepts light as shown in Figure 4.11a, and reflects it outwards over only the one

hemisphere facing the star. The result should be expressed in terms of the albedo, A, the

luminosity of the star, L?, and the distance of the planet to the star, r.

(b) Express Eq. (4.18) in terms of the emitted flux at the planet’s surface, Fem, instead of

its temperature.

(c) Determine the fraction of stellar flux that is reflected compared to that emitted

thermally, Frefl=Fem. For what value of albedo will the reflected flux equal the emitted

thermal flux?

10Neutron stars are usually detected as pulsars and it is the stronger non-thermal radiation (see Sect. 8.1), ratherthan their black body radiation, that is usually detected.


Appendix: derivation of the Planck Function

The derivation of the Planck Function (Ref. [76]) proceeds along the same principles as

we have already seen for the Maxwell–Boltzmann velocity distribution (Sect. 3.4.1),

the Boltzmann Equation (Sect. 3.4.5), and the Saha Equation (Sect. 3.4.6). That is, the

probability of finding a particle in any given excitation state depends on the statistical

weight (the number of possible ways the particle can fit into the state) as well as the

Boltzmann factor, eE=ðkTÞ. There are differences between this development and the

previous examples, however. One is that we are now considering ‘particles’ which are

photons. Photons are bosons (that is, particles with integer spin)11 that have two

possible spin states (þ1;1) corresponding to two opposite circular polarizations.

Photons do not have to obey the Pauli Exclusion Principle (see Appendix C.2) so the

limit to the number of possible states is set only by the Heisenberg Uncertainty

Principle (HUP, see below). Note also that, since photons are not bound particles

like electrons in an atom, we must consider the number of states in free space, or the

number of states per unit volume. Another difference is that the Boltzmann factor gives

a relative probability, for example, the number of particles in an energy state, a,

compared to an energy state, b. However, we now do not wish to compare the number

of photons in two particular states, but rather to determine the mean number of photons

or the mean energy in any state.

4.A.1 The statistical weight

To consider the number of energy states that are possible in a given volume, we can

equally ask how many momentum states per unit volume are possible, since E ¼ cp

(Table I.1), where p is the photon momentum and c is the speed of light. The answer is

dictated by the HUP which, for distinguishable states in three dimensions, can be

expressed as,

x px y py z pz h3 ð4:A:1Þ

The particle can be thought of as existing in a six-dimensional phase space in which

phase cells must be separated from each other by a cell of minimum size, (h3)12. The

number of possible phase cells is then given by the ratio,

x px y py z pz

h3ð4:A:2Þ

11Particles such as protons, neutrons, and electrons have half-integer spin.12The expression, dx dpx h

2, in Table I.1 is a statement of the uncertainty in the simultaneous measurement ofposition and momentum in one dimension. The expression given here uses the larger value, h, since this refers tohow far apart the phase cells must be to remain distinguishable.


Note that x y z represents a volume of real space and px py pz represents a

‘volume’ of momentum-space.

As was discussed for the Maxwell–Boltzmann velocity distribution (Sect. 3.4.1), the

momentum vectors of a particle can point in any direction so that any momentum

vector of magnitude between p and pþ dp, will end somewhere on a spherical shell

whose volume in momentum-space is 4p2dp. Thus, for this geometry, the number of

possible states (the statistical weight) is,

gðpÞdp ¼ V 8p2 dp

h3ð4:A:3Þ

where the additional factor of 2 is due to the fact that, for photons, the two different spin

states represent independent states.

Since we wish to derive a radiative spectrum, it is more useful to express this in terms

of frequency. Recalling (Table I.1) that p ¼ h=cðdp ¼ h d=cÞ, we finally have the

number of momentum (and therefore energy) states per unit volume,

gðÞdV

¼ 82d

c3ð4:A:4Þ

4.A.2 The mean energy per state

Now that we know the number of possible independent energy states per unit volume

for a photon in free space, we need to compute the number of photons we expect to be

in any given energy state.

Alhough, in principle, there is no limit to the number of photons per energy state, the

relative probability of finding n photons of energy, h , in a cell of frequency, , is given

by the Boltzmann factor, eðnh=kTÞ13. As indicated above, we do not want the relative

probability but rather the absolute probability, Pr, which is the relative probability

divided by the sum of all possible relative probabilities (similar to the computation, in

Sect. 3.4.5, of the partition function),

Pr ¼eðnh=kTÞPn e

ðnh=kTÞ ð4:A:5Þ

The mean number of photons per cell is then,

<N >¼P

n neðnh=kTÞP

n eðnh=kTÞ ð4:A:6Þ

13In fact, there is an additional 12h of energy in this factor that has been omitted here because it does not change

the development nor does it involve photons. It gives the energy of the vacuum.

APPENDIX: DERIVATION OF THE PLANCK FUNCTION 147

We now make the substitution, y ¼ ðhÞ=ðkTÞ and note that the denominator can be

expressed as,

Xn

eny ¼ 1 þ e1y þ e2y þ e3y þ . . . ¼ ð1 eyÞ1 ð4:A:7Þ

where we have used the binomial expansion of Eq. (A.2) using a ¼ 1; x ¼ ey, and

p ¼ 1. For the numerator, we need a sequence of expansions and factorizations,

Xn

neny ¼ 1e1y þ 2e2y þ 3e3y þ . . .

¼ e1y þ ðe2y þ e2yÞ þ ðe3y þ e3y þ e3yÞ þ . . .

¼ e1y ð1 þ e1y þ e2y þ e3y þ . . .Þþ e2y ð1 þ e1y þ e2y þ e3y þ . . .Þþ e3yð1 þ e1y þ e2y þ e3y þ . . .Þ þ . . .

¼ ð1 eyÞ1ðe1y þ e2y þ e3y þ . . .Þ¼ ð1 eyÞ1ðe1yÞð1 þ ey þ e2y þ . . .Þ

¼ ey

ð1 eyÞ2 ð4:A:8Þ

Substituting Eqs. (4.A.7) and (4.A.8) into Eq. (4.A.6) yields,

<N >¼ ey

ð1 eyÞ ð4:A:9Þ

Substituting back for y, dividing numerator and denominator by eh=kT, and multi-

plying by the energy, h, gives the mean energy per cell,

<E >¼ h

ðeh=kT 1Þð4:A:10Þ

4.A.3 The specific energy density and specific intensity

The photon energy per unit volume per unit frequency, or specific energy density

(Sect. 1.4), u , is the mean energy per state (Eq. 4.A.10) times the number of states per

unit volume (Eq. 4.A.4),

ud ¼ 8h3

c3

1

ðeh=kT 1Þd ð4:A:11Þ


Since the energy density, u , is related to the specific intensity, I , via,

u ¼4

cI ð4:A:12Þ

(Eq. 1.16 where, for isotropic radiation, I and J are equal), and defining the specific

intensity of a black body to be BðTÞ, we finally obtain,

BðTÞ ¼2h3

c2

1

ðeh=kT 1Þð4:A:13Þ

which is Planck’s law. The wavelength form is found by letting Bd ¼ Bd(see Eq. 1.5), and is,

BðTÞ ¼2hc2

5

1

ðehc=kT 1Þð4:A:14Þ

APPENDIX: DERIVATION OF THE PLANCK FUNCTION 149

PART IIIThe Signal Perturbed

An astronomical signal does not simply emerge from an object and travel through a

vacuum unperturbed. Rather, it may travel partially through the object that emitted it or

through interstellar clouds that could alter or extinguish it. It travels through space

which is itself expanding, and could pass near massive objects that bend its path. Thus,

we cannot hope to understand an astronomical object unless we first understand how its

signal may have become corrupted en route.

There is another way of looking at this, however. Every time the signal is changed, it

may be possible to learn something about the object or medium that produces the

change. Thus, an alteration in the signal is not always an unwanted nuisance, as was the

case for telescopes and (for astronomers at least) the atmosphere. Instead, the astro-

nomical signal itself might simply be a useful background source that probes the space

and matter through which it travels, like a flashlight illuminating the intervening

material for us. If the intervening matter neither emitted its own radiation nor had

any affect on a background signal (or nearby objects), it would be impossible to learn

anything about it.

Thus, we now focus more intently on astrophysical processes that can alter a signal

and what insights they might provide either about the signal itself or the altering

medium. We will first consider interactions in which the signal has contact with matter

(the interaction of light with matter) and then those interactions that are ‘non-touching’

(the interaction of light with space).

5The Interaction of Lightwith Matter

You see the motes all dancing, as the sun Streams through the shutters into a dark room. . . . Fromthis you can deduce that on a scale Oh, infinitely smaller, beyond your sight, Similar turbulence

whirls.

–the Roman poet, Lucretius, 1st Century B. C. (Ref. [102])

The interaction of light with matter spans the boundaries of classical and quantum

physics, and of microscopic andmacroscopic physics. One approach is to consider how

an individual photon or a wavefront might interact with an individual particle. Another

is to consider many particles together as representing a smooth medium that collec-

tively alters the intensity and/or the trajectory of an incoming wavefront. Both

approaches (and their variants) are valid and, if both exist, the resulting mathematical

description of the physical phenomenon must agree. In this chapter, we will consider

the interaction of light with matter in some detail, visiting each method along the way.

There are many types of matter that a photon could encounter in its flight across the

cosmos – hot ionized hydrogen gas, a cold cloud containing complex molecules, a dust

grain or even a planet. Such interactions with matter may produce a change in the

photon’s path and/or its energy. Although there are many types of matter, interactions

with individual particles fall into two categories, scattering and absorption. In the case

of absorption, a photon will disappear completely, its energy going into some other

form such as the thermal energy of the gas. For scattering, on the other hand, a photon

emerges from the interaction, though not necessarily at the same energy. The combined

effects of scattering and absorption, both of which remove photons from the line of

sight, are referred to as opacity (usually used for gases) or extinction (for dust).

The importance of an interaction with a particle depends on the cross-section

presented to the photon by the particle (e.g. Example 3.5) as well as the number


density of particles that the incoming light beam encounters as it travels through a

cloud. Therefore, in the next few sections, we focus on the scattering cross-section, s,

and the absorption cross-section, a (both in units of cm2) for the interaction of interest.

The dependence on particle density is considered later in Sect. 5.4.1. The special case

of dust grains is presented in Sect. 5.5.

5.1 The photon redirected – scattering

Scattering is a process by which the direction of an incoming photon changes as a result

of an interaction. As we will see, the interpretation of an interaction as ‘scattering’

rather than ‘absorption’ is sometimes unclear. This is because, at an atomic or

subatomic level, scattering involves the absorption and then re-emission of a photon.

However, scattering implies that the re-emission occurs ‘immediately’, meaning on a

short enough timescale that no other processes (e.g. collisions between particles) could

interfere and remove the energy supplied by the incident photon. Scattering can be

represented by the process: photon ! matter ! photon.

With a sufficient number of interactions, scattering tends to randomize the direction

of the photons. Thus, when an observer looks back along a line of sight, the signal is

attenuated (weaker) because many photons will be scattered into directions other than

the line of sight, as shown in Figure 5.1a. However, it is also possible for scattering to

be preferentially into a specific direction (symmetrically forwards and rearwards, for

example) depending on the properties of the scatterers and the wavelength of the

radiation. Reflection, for example, is just a special case of backwards-directed scatter-

ing, or ‘back-scattering’.

Depending on the geometry, scattering can also redirect light into the line of sight. One

example is when a dusty cloud surrounds a star. Photons that are emitted in directions

away from the line of sight will be scattered by the dust and a fraction of themwill scatter

in a direction towards the observer (Figure 5.2.a). A scattering dusty cloud around a

star is called a reflection nebula, an example of which is shown in Figure 5.3.

(a) (b)

Figure 5.1. Illustration of (a) random scattering and (b) absorption along a line of sight. The

observer is meant to be far from the source as well as the scattering medium in this diagram.

Darker arrows convey a more intense signal.

154 CH5 THE INTERACTION OF LIGHT WITH MATTER

A more spectacular example of scattering into the line of sight occurs when there is a

sudden increase in the brightness of an object, such as when a supernova (Sect. 3.3.3) or

Gamma Ray Burst (Sect. 5.1.2.2) goes off or when a star experiences a bright outburst as

can occur in certain types of variable stars. The light, travelling at c, scatters from

surrounding dust clouds at specific times after the event dependent on the distance to the

cloud. The result is called a light echo, an example of which is shown in Figure 5.4. A

more familiar example of light scattering into the line of sight occurs when the observer

is within the scattering medium itself, as is the case for the Earth’s atmosphere. Our sky

looks bright even in directions away from the direction of the Sun in the sky, because of

such scattering (Figure 5.2.b).

The probability that a photon will undergo a scattering interaction is proportional

to the scattering cross-section, s, a quantity that must be derived for each type

(a) (b)

Figure 5.2. Two examples of scattering into the line of sight. (a) Dust particles surrounding astar scatter its light randomly, but some of these photons will be directed towards the observer.Thus, the observer sees a faint glow around the star, much like the glow around a streetlight on amisty night. (b) When the observer is within a randomly scattering medium, light can enter his eyefrom all directions even though the original source of light is in one direction only.

Figure 5.3. The reflection nebula around the bright star, Merope, one of the stars in the Pleiadesstar cluster. The cross on the star is an artifact of the telescope and the other stars in the image arein the background or foreground. (Reproduced courtesy of Yuugi Kitahara, Astronomy Picture of theDay, March 1, 1999)

5.1 THE PHOTON REDIRECTED – SCATTERING 155

of scatterering process. Since a particle may present a different effective scattering

cross-section to photons of different wavelengths, this quantity is often wavelength-

dependent. Several scattering cross-sections and further details of some of the scatter-

ing processes that are important in astrophysics are given in Appendix D.We provide a

summary of them in the following subsections.

An important observational consequence of scattering, as explained in Appendix D

and illustrated in Figures D.1 and D.6, is that the scattered emission, depending on

viewing geometry, is polarized. The degree of polarization,Dp (introduced in Sect. 1.7),

is the fraction of the light that is polarized. If emission is observed at some position,

ðx; yÞ, on the sky and some fraction, Dp, is polarized, then this fraction of the total

intensity may have been generated at a different location and then scattered by particles

at ðx; yÞ into the line of sight. Some care must be exercised in this interpretation,

Figure 5.4. The X-ray ‘afterglow’ of the Gamma-Ray Burst source, GRB 031203, is seen at thecentre of these images. The GRB was detected in December/2003 and is in a distant galaxy. The fourX-ray images, taken from the XMM-Newton satellite, span the interval from 25 000 s (6.9 h) to55 000 s (15.3 h) after the GRB was first detected. The rings are caused by X-rays scattering fromdifferent dust screens in our own Milky Way that are illuminated sequentially. The larger rings areseen at later times because it takes longer for light to travel from the GRB over the large angle andthen to the observer, in comparison to light travelling closer to the GRB line of sight. The imagesgive the appearance of an expanding ring, but are only produced by dust reflections. This is the firstobservation of a light echo seen at X-ray wavelengths. (Reproduced by permission of SimonVaughan, Univ. Leicester)


however, because, as indicated in Sect. 1.7, polarized emission can also result from

certain processes intrinsic to the energy generation mechanism. Generally, other

available information about the source is also considered in order to decide whether

or not the polarization is due to scattering1.

5.1.1 Elastic scattering

If there is no change in the energy of the photon as a result of the interaction, then the

scattering is said to be elastic, analogous to a small particle bouncing off a wall without

losing kinetic energy. If light is treated as a wave, then such interactions are called

coherent, meaning that the outgoing energy, and therefore wavelength (recall

E ¼ hc=) has not changed from the incoming one. Some examples follow.

5.1.1.1 Thomson scattering from free electrons

Thomson scattering (Appendix D.1.1) is the elastic scattering of light from free

electrons in an ionized or partially ionized gas. The scattering is elastic provided the

energy of the incoming photon is much less than the rest mass energy of the electron,

Eph mec2. Otherwise, some energy would be imparted to the electron (see Sect.

5.1.2.2 for that case). Thomson scattering is a very common process in astrophysics

and occurs wherever ionized gas is found2, from regions as diverse as stellar interiors

(see Example 3.5) to the intergalactic medium.

A characteristic of Thomson scattering is that the scattering cross-section is inde-

pendent of wavelength (i.e. it is ‘grey’, see Eq. D.2). This means that a radio photon is

equally as likely to be scattered as an optical or UV photon. Thus, if the degree of

polarization, Dp, is independent of wavelength, it is evidence that Thomson scattering

may be occurring. This appears to be the case for the nuclear region of the galaxy, NGC

1068 (see Figure 5.5). This galaxy’s active galactic nucleus (AGN) is obscured from

direct view, but free electrons in the dense ionized gas around the nucleus scatter AGN

photons into the line of sight. Moreover, since the scattering is independent of , thescattered spectrum looks like the spectrum of the AGN. Thus the ionized gas acts like a

mirror that allows us to see into the AGN itself. This is a good example of how a process

that might be thought of as ‘corrupting’ the signal has actually turned into an indis-

pensable tool for studying one of the most powerful types of objects in the Universe

(see Figure 1.4 for another example of an AGN).

1Examples are the nature and waveband of the spectrum. For example, if polarization is seen at radiowavelengths,some or all of the emission might be due to synchrotron radiation (Sect. 8.5).2Wherever there is ionized gas there will also be free protons. However, the proton scattering cross-section isnegligible in comparison to the electron scattering cross-section (see Appendix D.1.1 and Eq. D.2).


5.1.1.2 Resonance (line or bound–bound) scattering

When the frequency of incoming light matches or is close to the frequency of an energy

transition of a bound electron in an atom or molecule, the electron responds strongly

and in resonance with the forcing of the wave’s oscillating electromagnetic field,

as outlined in Appendix D.1.2. This is called resonance scattering and it is coherent

when the bound electron is excited to an upper state by absorbing the photon and then

de-excites to the initial level again, re-emitting a photon at an equivalent energy3 in

Figure 5.5. (a) The galaxy, NGC 1068, belongs to a class of galaxy that has a very active galacticnucleus (AGN), most likely due to a supermassive black hole. (Reproduced courtesy of Pal. Obs.DSS). (b) Highly energetic activity around the black hole creates outflow which is collimated (withthe help of a circumnuclear disk) into a bipolar radio jet, shown here at 73 cm (Ref. [120]).(Reproduced courtesy of the NASA/IPAC Extragalactic Database). (c) A high resolution UV image isshown here in grey scale (Ref [88]). The AGN is marked, and the vectors show the orientation of theelectric field of Thomson scattered UV emission from the AGN. The fraction of the emission at anyposition that is polarized is actually scattered AGN emission. The measured degree of polarization,Dp, ranges from 10 per cent to 44 per cent (Eq. 1.34), depending on position. An image at anotherwaveband should show the same value of Dp at any given position in the case of Thomsonscattering. (Reproduced by permission of M. Kishimoto). (d) A possible model for Thomsonscattering of AGN emission (Ref. [58]). The AGN itself is obscured from direct view but electrons inthe ionized region scatter the emission into the line of sight. (Reproduced by permission of M. Elvis)

3The emitted photon may have a slightly different energy since the emission probability is distributed infrequency according to the Lorentz profile (Figure D.2). Other effects will broaden the line even more (see Sect.9.3) so coherence is not perfect but the shift in energy is small in comparison to the energy of the line itself.


another direction. As with Thomson scattering, resonance scattering results in a

polarized signal (Ref. [154]).

An example of a resonance scattering line is the Ly line (a transition between

quantum number n ¼ 1 and n ¼ 2) of hydrogen. This is a UV transition at a rest

wavelength of 122 nm with an upper state lifetime of only 109 s (Table C.1). Since

this is a much shorter timescale than the time between collisions under most interstellar

circumstances (see Example 5.1), incident UV radiation near the Ly wavelength is

strongly scattered in this line.

Example 5.1

A Ly photon travels through a pure hydrogen gas of density, n ¼ 103 cm3 andtemperature, T ¼ 104 K. If the gas is mostly ionized and the photon excites an upwardsLy transition in one of the small fraction of particles that are neutral and in the groundstate, will the outcome be scattering or absorption? Consider the permitted 2p ! 1stransition only.

Once the Ly photon has excited the atom, the electron will spend a time, 2;1 ¼ 1A2;1

¼1:6 109 s (Eq. D.22, Table C.1 plus notes to table) in the energy level, n ¼ 2 before

spontaneously de-exciting to the ground state, n ¼ 1, again. In the mean time, there are

collisions between particles and there is a possibility that de-excitation will occur via a

collision with a free electron4 or that collisional excitation will occur to higher levels.

Considering only downwards permitted transitions, from Table 3.2 the relevant collision

rate coefficient for de-excitation of this transition at the given temperature is

Lyð2p!1sÞ ¼ 6:8 109 cm3 s1 so the timescale between de-exciting collisions is

(Eq. 3.19) t ¼ 1=½ð103Þð6:8 109Þ ¼ 1:5 105 s. Since 2;1 t, the de-excitation

will occur spontaneously. Similarly, it can be shown that the downwards collisional

timescale via the 2s ! 1s transition, as well as upwards collisional timescales are longer

than the spontaneous de-excitation timescale under the conditions described (see also Sect.

8.4). Once spontaneous de-excitation occurs, the re-emitted Ly photon (after having

travelled a mean free path) can excite another particle, followed by spontaneous

de-excitation again, and so on. Thus, the photon will be scattered5.

The Ly photon scattering cross-section, as developed for resonance scattering

in Appendix D.1.4, is given in Table 5.1. However, it is important to note that

thermal motions of many atoms in a gas (see Sect. 9.3) will result in a line that is

much wider in frequency and has a lower peak scattering cross-section than a line

from a single isolated atom at rest. Therefore, a sample cross-section for 104 K gas

is also given in Table 5.1. Even when one considers a more realistic line width, it is

4Collisions with protons are less likely since protons are massive and slow moving in comparison to electrons.5Note that we have neglected the possibility of photo-absorption from the n ¼ 2 level to a higher level, but thiswill only be important in a strong radiation field.


clear that this scattering cross-section is quite high in comparison to other values in

the table.

HII regions (e.g. Figures 3.13, 8.4) are good examples of objects in which Ly resonance scattering is important. In HII regions, photons at the wavelength of Ly are numerous, both from Ly recombination line emission as well as the continuum

of the central ionizing star. As explained in Sect. 3.4.7, an HII region is almost

entirely ionized but, at any time, recombinations are occurring and some atoms will

have electrons in the ground state. Therefore, once a Ly photon is emitted, it will

be reabsorbed by other atoms, exciting the electron into the state, n ¼ 2, followed by

re-emission to the ground state again, and so on. There will be many such scatterings

within the nebula because, not only is the resonance scattering cross-section large

(Table 5.1), but also the densities of HII regions are low, minimizing the possibility

that a collision could interfere with the scattering process. Even though an HII

region has a very low density in comparison to the interior of a star (see Table 3.1),

Ly photons in an HII region take a random walk out of the nebula much like

we saw for the interior of the Sun (see Sect. 3.4.4). Because Ly photons

are scattered so strongly, the nebula is actually opaque (see Figure 4.2) at this

Table 5.1. Sample photon interaction cross-sectionsa

Wavelength Cross-section

Type Description or energy (cm2)

Tb Thomson scattering 0:51MeV 6:65 1025

KNc Compton scattering 0.51 MeV 2:86 1025

5.1 MeV 8:16 1026

Rd Rayleigh scattering (N2) 532 nm 5:10 1027

(CO) 532 nm 6:19 1027

(CO2) 532 nm 12:4 1027

(CH4) 532 nm 12:47 1027

bbe Ly (natural)f 121.567 nm 7:1 1011

Ly ð104 KÞg 121.567 nm 5:0 1014

HI!HIIh H ionization 13.6 eV 6:3 1018

ffi free-free absorption 21 cm 2:8 1027

aCross-sections apply to a single scattering event from a single particle.bThomson cross-section (Eq. D.2).cKlein–Nishina cross-section for Compton scattering (see Figure D.4).dRayleigh scattering cross-section for a temperature of 15C and pressure of 101 325 Pa(Ref. [157]).eResonance, bound–bound scattering from the line centre.fFrom the natural line shape using Eq. (D.12) with data from Table C.1. Note that only thepermitted transitions have been included (see notes to Table C.1).gAs in the previous row but assuming that the line is Doppler broadened (see Sect. 9.3) atthe temperature indicated (Ref. [144]).hPhotoionization cross-section from the ground state (Eq. C.9).iFree–free absorption cross-section for the conditions: ne ¼ 0:1 cm3 and T ¼ 104 K. Thecross-section will vary with these quantities and also decreases with increasing frequency(Ref. [160]).


wavelength6 (Example 5.2). Other Lyman lines are similarly scattered, though to a

lesser extent than Ly .

Example 5.2

What is the mean free path for scattering of a Ly photon in an HII region of density,n ¼ 100 cm3?

HII regions are typically at a temperature of T ¼ 104 K (Table 3.1). The scattering cross-

section is bb ¼ 5:0 1014 cm2 at this temperature (Table 5.1) so the mean free path of a

photon is,

l ¼ 1

nbb

¼ 1

ð100Þ ð5:0 1014Þ ¼ 2 1011 cm ð5:1Þ

This is less than three times the radius of the Sun and is a very short distance in

comparison to a typical HII region which can be hundreds of parsecs in size. Therefore,

the Ly photon will take a random walk in an HII region in order to escape.

Another example of Ly resonance scattering can be seen in the Ly forest, an

example of which is shown in Figure 5.6 (bottom). In the spectra of distant QSOs (Sect.

1.6.4), many lines can be seen on the shorter wavelength (blue-ward) side of the Ly line that is emitted from the QSO itself. These lines result from absorption and

re-emission (resonance scattering) from the Ly line in many individual intergalactic

HI clouds, each at a different distance along the line of sight towards the QSO. (The

differing distances correspond to different redshifts and therefore frequencies along the

line of sight, as described in Sect. 7.2.) The Ly forest lines are commonly referred to

as ‘absorption lines’ because what the observer actually sees is removal of the back-

ground QSO light. However, the process is one of resonance scattering since, at the

densities of these clouds (see Table 3.1), an atom that has absorbed a photon will re-

emit in a random direction before it will collide with another particle. The HI clouds

that produce most of the observed Ly forest lines presumably permeate space but are

too faint to be observed directly in emission with current telescopes. They are only seen

in absorption in front of bright background sources. However, numerical simulations

suggest that they form a cosmic web similar to that shown in Figure 5.7.

Although hydrogen is the most abundant atom, other elements are present in

astrophysical sources and many of these atoms also have resonance lines (see

Ref. [154] for examples). They can also play an important role in the transfer of

radiation through a source.

6The situation in which downward transitions to the n ¼ 1 level are immediately re-absorbed in HII regions iscalled Case B. Case A assumes that all such emission escapes from the nebula without any interaction. However,the density of a Case A nebula would have to be so low that it would be too faint to be detectable. Case B isconsidered to be the most realistic situation (Ref. [115]). See also Sect. 9.4.2.


Figure 5.6. Spectra of the quasars, 3C273 (top) and Q1422þ2309 (bottom). The quasar, 3C273 isrelatively nearby at a redshift of z ¼ 0:158 (see Sect. 7.2.2 for a discussion of the expansionredshift) whereas Q1422þ2309 is much farther away. The spectra have been aligned so that the Ly line emitted from the quasars, themselves, is shown at its rest wavelength. In between us and themore distant quasar, Q1422þ2309, there are many HI clouds at a variety of distances, creating themany absorption lines seen to the left of the quasar emission. This is called the Lyman alpha forest.Most intergalactic gas is, however, ionized. (Reproduced by permission of William Keel)

Figure 5.7. One view of the cosmic web from the ‘Millennium Simulation’ which used 1010

particles in a computer code to simulate the formation of structure in the Universe (Ref. [161]). Theimage represents the expected structure of dark matter (see Sect. 3.2) but the structure of the Ly forest may be similar. This image is for a redshift of z ¼ 1:4 (see Sect. 7.2.2), corresponding to atime, 4.7 Gyr after the Big Bang and the unitless quantity, h ¼ H0=ð100 kms1 Mpc1Þ (seeAppendix F for information on H0). (Reproduced by permission of V. Springel and the VirgoConsortium)


5.1.1.3 Rayleigh scattering and the blue sky

The most famous example of Rayleigh scattering is the scattering in the Earth’s

atmosphere leading to the blue colour of the sky. It was Leonardo da Vinci who,

around 1500, first suspected the reason for the blue sky, in particular by his observation

that wood smoke looked blue when observed against a dark background (Ref. [79]).

The effect was finally described quantitatively in the year 1899 by Lord Rayleigh

whose name is now associated with the phenomenon.

Rayleigh scattering occurs when the incoming signal has a wavelength, , that ismuch larger than the resonancewavelengths of a bound electron in an atom ormolecule

(see Appendix D.1.2). For optical light impinging on particles with UV transitions, this

also means that the size of the scattering particle, as discussed in Appendix D.1.5.

Because of the strong dependence of the scattering cross-section on wavelength,

s / 1=4 (see Eq. D.10, D.13, or D.24), shorter wavelength blue light7 will scatter

more efficiently than longer wavelength red light. Blue light has a wavelength of

470 nm (Table G.5) and, since the most abundant molecules in the atmosphere,

namely nitrogen and oxygen, are about 0.3 nm in size, atmospheric scattering is firmly

in the Rayleigh regime. Small dust particles also contribute, but the dominant scatter-

ing is due to molecules and the sky would still look blue even without the additional

contribution by dust.

For a geometry as shown in Figure 5.2b, blue light is more likely to scatter into the

line of sight of the observer than red light. Thus the yellow sun produces a blue sky to

an observer on the Earth. Though less obvious, the nighttime sky is also blue. Although

the feebleness of the light at night makes it impossible to discern by eye, a time

exposure reveals the colour, as Figure 5.8 shows.

If therewere no atmosphere, the daytime sky would be black except at the position of

the Sun itself. The fact that the clear sky atmosphere is transparent (Example 5.3)

means that most photons travel through it unimpeded with only a small fraction

scattered. This is why, on a clear day, the brightness of the Sun is much greater than

the brightness of the blue sky.

Example 5.3

What is the mean free path of an optical photon to Rayleigh scattering in the Earth’satmosphere?

The mass density of the atmosphere at sea level (the densest air) is 1:2103 gm cm3. The mean molecular weight of air is m ¼ 28:97 so the number density of

air at sea level is (Eq. 3.11), n ¼ =ðmmHÞ 2:5 1019 cm3, where mH is the mass of a

hydrogen atom. The most likely interaction is with the most abundant molecule, N2, so its

Rayleigh scattering cross-section at 532 nm (Table 5.1) together with Eq. (3.18) yields,

7Violet light is scattered even more but our eyes are less sensitive to violet than blue.


l ¼ 1=ðnRÞ 80 105 cm. At higher altitude, the density decreases, so the mean free

path increases. Thus, l > 80 km. Recall that this lower limit is well into the ionosphere

(Sect. 2.3), confirming what we already know – that the clear sky is optically transparent.

For a geometry as shown in Figure 5.1a, blue light is more likely to be scattered out

of the line of sight than red light. Thus any light-emitting object above the Earth’s

atmosphere will be reddened (as first introduced in Sect. 2.3.5) and also dimmer, due to

Rayleigh scattering. The Sun is redder than its true colour even when it is high

overhead. If the line of sight through the atmosphere is longer, such as when viewing

the Sun at sunrise or sunset, (see Figure 2.13), then the redder colour is more enhanced

andmore obvious to the eye8. The same effect can be observed for other objects such as

the Moon, planets or stars. Individual scattered photons, however, have the same

wavelength as the incident photons, so even though Rayleigh scattering is wavelength

dependent, it is still an example of elastic scattering.

Rayleigh scattering produces polarized light in the same way as Thomson scattering

(see Figure D.1). Even though the Sun emits light that is unpolarized, for example, its

Figure 5.8. A time exposure (ISO 200, f/2.8, 69 s) reveals the blue colour of the night sky in thisun-retouched image. (Reproduced by permission of Joseph A. Shaw, Montana State UniversityBozeman, Montana). (see colour plate)

8Scattering (and some absorption) from dust, water vapour and large molecules can also contribute to reddening(see also Prob. 6.1).


scattered light will be polarized at a 90 viewing angle, as can be verified by looking

near the horizon with a polarizing filter when the Sun is overhead. Also, like Thomson

scattering, Rayleigh scattering provides a way of looking at a source via its ‘reflection’,

although weighted by the 1=4 wavelength dependence. It is thus possible to see the

Sun’s (weighted) spectrum by pointing a spectrometer at some position in the sky away

from the Sun itself. The Sun’s Fraunhofer lines9, for example, can easily be seen in

such a fashion. The faint optical light of a reflection nebula (see Figure. 5.3) will also be

polarized.

On a final note, the blue sky of the atmosphere can be contrasted with the greyer

colour associated with water droplets in clouds. Since water droplets are not small

with respect to the wavelength of light, scattering from these particles is not in the

Rayleigh scattering regime. The wavelength dependence of large particle scattering is

flatter than for Rayleigh scattering, hence the greyish colour of clouds, as described in

Appendix D.3.

5.1.2 Inelastic scattering

If the scattered photon is of lower energy (longer ) than the incident photon, then thescattering is said to be inelastic or incoherent. Several examples follow.

5.1.2.1 Bound–bound inelastic scattering and fluorescence

Inelastic scattering can occur from the bound–bound excitation of an atom to a high

energy state followed by cascading to other energy levels between the high state and

the initial one. Thus, a high energy photon is degraded into multiple lower energy

photons. The emission of light in this cascading process is called fluorescence.

There are various possible outcomes for bound–bound scattering, depending on the

incident photon energy and the final state of the atom. If the final energy state is the

same as the initial one, then the sum of the energies of the emitted photons equals

the energy of the incident photon, and the situation is one of pure inelastic scattering

(Prob. 5.3). If the final state of the atom is an excited state higher than the initial state,

then some energy is lost to the excitation of the atom. Eventually, an atom will always

decay spontaneously from any excited state to its ground state but, if an energy-

exchanging collision occurs in the mean time, then the energy difference will be lost in

the collision. In this case, some of the incident photon energy is turned into lower

energy photons and some goes into heating the gas.When energy is ‘lost’ via excitation

or heating, the resulting process is called absorption (further considered in Sect. 5.2).

The requirement that scattering be inelastic, however, refers only to a comparison

between the incoming and outgoing photon energies.

9The Sun’s Fraunhofer lines are the absorption lines formed in the Sun’s photosphere, as described in Sect. 6.4.2.See also Figure 6.4.


5.1.2.2 Compton scattering at high energies

Compton scattering (see Appendix D.2) is the scattering of a photon from a free

electron in the high energy limit, i.e. when the energy of the incident photon is of order

or greater than the rest mass energy of the electron, Eph0mec2 ¼ 0:51 MeV. In

Compton scattering, some of the photon energy is imparted to the electron giving it

a velocity, as illustrated by Figure D.3. The energy of the scattered photon,

E0ph ¼ hc=0, is less than the energy of the incident photon (Eq. D.26), the difference

going into the kinetic energy of the electron. The result is the cooling of the radiation

field, the scattered photons having longer wavelengths due to loss of energy.

Since 0.51 MeV is in the gamma-ray part of the spectrum (Table G.6), Compton

scattering is associated with highly energetic objects, such as active galactic nuclei

(AGN), binary stars in which at least one member is a compact object such as a neutron

Figure 5.9. The optical afterglow of the powerful gamma-ray burst, GRB 990123, seen here as abright point source, was caught by the Hubble Space Telescope in this image. The fact that this GRBwas found to be associated with a galaxy (faint extended emission) provides powerful evidence thatGRBs are located in distant galaxies previously too faint to have been observed. (Reproduced bypermission of the HST/GRB collaboration/NASA). The inset shows an artist’s concept of anextremely massive star beginning its gravitational collapse with light at gamma-ray energies beingbeamed into two jets. Compton scattering, among other processes, takes place in the initial phasesof the GRB. (Reproduced by permission of NASA/SkyWorks Digital)


star, or in regions around black holes. Other highly energetic sources are the transient

gamma-ray bursts (GRBs) such as introduced in the light echo example of Figure 5.4.

These are bursts of gamma-ray emission of very short duration, from milliseconds to

2 s for the short duration bursts, and from 2 s to several minutes for the long duration

bursts. Discovered by spy satellites in the 1960s, GRBs were found to be associated

with distant faint galaxies in the 1990s (see Figure 5.9). GRBs represent the most

powerful explosions in the Universe and can be 100 times more powerful than super-

novae (Sect. 3.3.3). The long duration events may be related to hypernovae – very

powerful supernovae that originate from precursor stars more than 20 times the mass of

our Sun. A possible precursor is a Wolf–Rayet star which is known to have high mass

loss via stellar winds (see Figure 5.10). The interaction of the hypernova with wind

material may also play a part in forming the GRB. The short duration GRBs likely have

a different origin and may be associated with colliding neutron stars or collisions

between a neutron star and black hole. It is difficult to explain the high energies of

GRBs without requiring that the detected emission be beamed in our direction, as

Figure 5.9 suggests (see also Prob. 1.4).

Compton scattering is the dominant scattering process for gamma-ray photons in the

range, 0.51 to a few MeV, which is the low energy part of the gamma-ray spectrum. In

Figure 5.10. The 2.5 pc diameter nebula, M1-67, shown in this H image from the HubbleSpace Telescope, is caused by a massive stellar wind from the Wolf–Rayet star seen at the centre.Wolf–Rayet stars are examples of the youngest, hottest, and most massive stars known and arelosing mass via stellar winds at rates of 105104M yr1. This nebula has a mass of 1:7M andan age of no more than 104 yr, consistent with a very high mass loss rate (Ref. [69]). (Reproducedby permission of Y. Grosdidier et al., U. of Montreal, 1998, ApJ, 506, L127, and NASA)


this regime, the cross-section decreases strongly with increasing photon energy,

ensuring that the probability of interaction is low. Thus, gamma-rays in this energy

regime are quite penetrating and strong effects are only observed in regions of high

density, such as near the source itself (Prob. 5.8). At higher energies, other processes

also become important, such as the conversion of a gamma-ray into an electron/

positron10 pair, or the interaction of the gamma-ray with atomic nuclei or even with

other photons. Consequently, modelling a highly energetic source, like a GRB, requires

that a variety of processes be considered (e.g. Figure 10.1).

Appendix D.2 provides an expression for the energy of the scattered photon when

the electron is initially at rest. However, a more general development takes into

account the initial kinetic energy of the electron and reveals that the inverse energy

exchange can also occur. That is, if the electron is more energetic than the photon

(see Eq. D.29 for more exact condition), then the electron can impart an energy to

the photon, a process called inverse Compton scattering which will be discussed

further in Sect. 8.6.

5.2 The photon lost – absorption

Absorption occurs when a photon loses all of its energy (disappears) because of an

interaction. In Sect. 5.1.2.1, we gave an example of how a fraction of the energy of an

incident photon could be absorbed if a bound–bound absorption occurs, followed by

radiative de-excitations to a higher energy state than the initial one. True absorption,

however, requires that there is no scattered photon at all; the entire incoming photon

energy is lost. Following the symbolism of Sect. 5.1, absorption can be represented as:

photon ! matter. Like the scattering cross-section, the absorption cross-section refers

to the effective cross-sectional area that a particle presents to an incoming photon for an

absorptive process. If a photon is absorbed, energy must still be conserved, so the

incoming photon’s energy must be converted into some other form. There are primarily

two possibilities for the absorption of light by matter: kinetic energy gain (which

includes heating), or a change of state.

5.2.1 Particle kinetic energy – heating

If a photon is absorbed by matter, imparting a motion to it, then the photon energy

ðE ¼ hÞ is converted into kinetic energy ðE ¼ 12m2Þ. This occurred for Compton

scattering (Sect. 5.1.2.2) but, in that case, a scattered photon remained so the process

was categorized as inelastic scattering. We also saw this when we considered radiation

pressure (Sect. 1.5). A photon that was completely absorbed by a surface imparted a

momentum to that surface. With a sufficiently powerful beam, the surface could be

10A positron is the anti-particle to an electron. It has the same mass but a positive charge.


pushed along like a sail. Some of the absorbed photons, however, will contribute to

heating the sail. For matter in any phase, heat involves the random internal motions of

its constituents and is therefore a form of kinetic energy.

For solids, such as the Solar sail, a dust grain, or a planet, these motions refer to

vibrations of the particles within their internal crystalline structure. As was described

in Sect. 4.2.1, a dust grain, asteroid or planet that does not move from its position with

respect to the light source will eventually achieve and maintain an equilibrium

temperature as a result. For dust, it is only the larger grains that actually maintain an

equilibrium temperature, since VSGs (see Sect. 3.5.2) cannot retain internal heat and

tend to lose it quickly before another photon can be absorbed. The interaction of light

with dust is a more complex process and will be considered separately in Sect. 5.5.

If the photon impinges on a gas, then it can contribute to the kinetic energy of

individual particles in the gas. Since the randomly moving gas particles share energy

via collisions (Sect. 3.4.1), any absorption process that is immediately followed by an

energy-exchanging collision will result in the loss of the photon. An example is if a

particle experiences a bound–bound excitation followed immediately by a collision

(the opposite conclusion to Example 5.1).

In an ionized gas, excess energy from the ionization process goes into the electron

kinetic energy (see next section) that is then shared via collisions. Another important

absorption process in ionized gases occurs when a free electron is in the vicinity of a

positive nucleus so that it feels a small force by the nucleus as it travels past. In such a

case, a photon can be absorbed by the electron, giving the electron a higher kinetic

energy and different trajectory than it originally had. This is called free–free absorp-

tion11. The free–free cross-section depends on frequency, temperature and density. A

sample value is given in Table 5.1.

5.2.2 Change of state – ionization and the Stromgren sphere

Absorbed photons can also produce a change of state, such as the transition from a solid

to a gas (sublimation), from a neutral gas to an ionized gas (ionization), or from a

molecular gas to a neutral atomic gas (dissociation). These processes could also occur

via particle collisions of sufficient energy, so terms such as photoionization are some-

times used to distinguish the photo-absorptive process from collisional ionization.

Photoionization, in particular, is very common in astrophysics and will occur anywhere

that photons are energetic enough to ionize the neutral material that may be nearby. It is

useful to remember that there are many constituent elements in the Universe and each has

its own ionization energy. These elements can be very important in terms of the heating

and cooling balance in various objects (neutral and singly ionized carbon in the ISM is a

good example). For the most abundant element, hydrogen, the required energy for

ionization from the ground state (its ionization potential) is 13.6 eV (Eq. C.5) which is

11The inverse of this effect, free–free emission, is discussed in Sect. 8.2

5.2 THE PHOTON LOST – ABSORPTION 169

in the UV part of the spectrum (Table G.6). Any photon of equivalent or greater energy, if

absorbed by the atom, would ionize it and any excess energy would go into the kinetic

energy of the ejected electron, contributing to heating of the ionized gas. The photo-

ionization cross-section for hydrogen in its ground state is given in Table 5.1.

Photoionization occurs in a variety of astronomical objects. Some examples are

planetary nebulae (Figure 3.7) in which the source of ionizing photons is the central

white dwarf, the outer gaseous ‘edges’ of the disks of galaxies for which the source of

ionizing photons is the weak extra-galactic radiation field, the regions around

active galactic nuclei (e.g. Figure 5.5) ionized by the AGN itself, and HII regions

(Figures 3.13, 8.4) which are ionized by a hot central star or stars. Even the re-

ionization of the Universe itself after the ‘dark ages’ (see Sect. 1.4) is described by

this process, although it is not yet clear whether the re-ionization was caused by the first

stars or the first AGNs (Prob. 5.10). The quest to understand which sources first ‘turned

on’ in the Universe is currently of much interest and the time period over which this

occurred is referred to as the Epoch of Re-ionization (EOR).

If an ionized region is not expanding or shrinking and the degree of ionization (the

fraction of particles that are in an ionized state) is constant, then the ionization rate

must equal the recombination rate and we say that the region is in ionization equili-

brium. Such an argument is similar to the one used for temperature equilibrium in Sect.

4.2.1. This fact can then be used to determine some physical parameters of the region.

In Example 5.4, we do this for the simple case of an HII region around a single star. The

star must be a hot, massive, O or B star since only these stars have a sufficient number

of UV photons to form HII regions (e.g. Prob. 5.9). In an HII region, the gas is almost

completely ionized and we assume that every photon of sufficient energy leaving the

star will eventually encounter a neutral atom that it then ionizes. Such a development

was first presented by B. Stromgren, and so an HII region around a single star is

sometimes called a Stromgren sphere which has a radius called the Stromgren radius.

Example 5.4

Find the Stromgren radius of an HII region consisting of pure hydrogen of uniformelectron density, ne ¼ 1000 cm3 and temperature, T ¼ 104 K around a single central O8Vstar.

As described in the text, Ni ¼ Nr, where Ni is the number of ionizing photons per second

leaving the star and Nr is the number of recombinations per second within the nebula.

Beginning with the recombination rate, a recombination can occur whenever an electron

‘collides with’ a proton. We use the total collision rate per unit volume, tot from Eq. (3.2.1)

multiplied by the total volume, V of the HII region and use the collision rate coefficient

appropriate to recombination, called the total recombination coefficient r (Table 3.2),

Nr ¼ tot V ¼ ne np r V ¼ n2e r

4

3RS

3 ð5:2Þ


where np is the proton density, taken to be equivalent to ne for pure ionized hydrogen

and RS is the Stromgren radius. Equating Nr to Ni and rearranging we find,

RS ne2=3 ¼ 3

4

Ni

r

1=3

ð5:3Þ

(all cgs units). The RHS of this equation contains information about the exciting star

(plus constants12) and the LHS contains information about the HII region, often

combined into a single parameter called the excitation parameter, U,

U RSne2=3 ð5:4Þ

The excitation parameter has cgs units of cm cm2 but is often expressed in units of pc

cm2 (see Sect. 8.2.2, for example). If the exciting star is known, implying that Ni is

known (see below), Eq. (5.3) relates the size of the HII region to its density. If either

quantity is known, the other can be calculated, although a more accurate derivation

would consider non-uniform densities and also take into account some absorption of

photons by other elements such as helium, or by dust particles.The quantity, Ni, can be computed from the ionizing luminosity, Li (erg s

1), of the star

divided by the photon energy, h (erg). The photon energy, however, can have a variety of

values provided it is greater than the ionization potential, that is, i, where i is thethreshold frequency required to ionize hydrogen. Li can be determined from an integration

of the Planck curve, BðTÞ (Eq. 4.1), using Eqs. (1.10) and (1.14),

Ni ¼Li

h ðiÞ¼ 42 R2

Z 1

i

BðTÞh

d ð5:5Þ

where R is the radius of the star. For an O8V star, T ¼ 37 000 K and R ¼ 10R ¼6:96 1011 cm (Table G.7). Numerical integration with i ¼ 3:28 1015 Hz yields

Ni ¼ 7:95 1048 photons s1. Using this value in Eq. (5.3) along with ne ¼ 1000 and

r ¼ 2:59 1013 cm3 s1 (Table 3.2), gives RS ¼ 1:93 1018 cm or 0.63 pc.

As indicated in the above example, this simple development links the stellar type to

the size and density of the HII region. If the stellar type is known and the density

determined by other means (e.g. Sect. 8.2.2) the angular size can be compared to RS to

find the distance. HII regions have distinct boundaries (Example 5.5) so this measure-

ment is straightforward. If the distance is known, the size can be used to determine the

kind of star that may be ionizing the nebula or whether more than one star is required.

Although corrections are needed for dust and density variations, the simple result of

Eq. (5.3) can still be very useful in placing constraints on the quantities of interest.

12The recombination coefficient is not exactly constant but varies only slowly with temperature.

5.2 THE PHOTON LOST – ABSORPTION 171

Moreover, the arguments apply to any ionized region. If the source of ionizing radiation

is an AGN at the centre of a galaxy, then one need only insert the spectrum of an AGN

(a power law) instead of the Planck function in Eq. (5.5).

Example 5.5

Why do HII regions and planetary nebulae have sharp boundaries?

If we assume a density of n ¼ 100 cm3 (Table 3.1) and use the cross-section for

photoionization in Table 5.1, the mean free path (Eq. 3.18) of a photon to the ionization

of hydrogen is l ¼ 1=ðnHI!HIIÞ ¼ 1=½ð100Þð6:3 1018Þ ¼ 1:6 1015 cm 100AU.

Thus, the transition region between an ionized and a neutral gas of this density is about

100 AU. Assuming a spatial resolution of 100 (Sect. 2.3.2) is possible, the HII region or

planetary nebula would have to be closer than 100 pc (Eq. B.1) in order for this region to be

spatially resolved. The closest known HII region is the Orion Nebula at a distance of 460 pc

and the closest planetary nebula is likely the Helix Nebula at a distance of about 140 pc.

Therefore all HII regions and planetary nebulae have boundaries that are sharp, that is, they

cannot be resolved.

5.3 The wavefront redirected – refraction

Refraction is another example of light interacting with matter. As light passes from one

kind of transparent medium to another with a different index of refraction, n, the

wavefront changes direction according to Snell’s law (Table 1.1). An example that

we have already seen is the systematic bending of light as it enters the Earth’s

atmosphere (see Figure 2.A.1 in the Appendix at the end of Chapter 2). Figure 5.11

shows how the bending of a wavefront can be thought of as the result of interactions

that slow the speed of light, , through the mediumwith the higher index of refraction13

(from Table I.1, n ¼ c=). Although other subtleties are involved, this illustration is

useful in providing a visualization of the refraction phenomenon from both a wave and

particle viewpoint.

Aside from the Earth’s atmosphere, refraction can also occur under other astro-

physical conditions. For example, two media of the same composition but a different

density should have different indices of refraction. Thus, in principle, any gas within

which density variations exist should also experience some refractive effects. For

neutral gas, the degree of bending turns out to be quite negligible. However, refraction

can become important when a signal propagates through an ionized gas (a plasma, see

Appendix E) or through a neutral gas that has a significant fractional ionization.

13A similar decrease in the speed of light in a medium was considered for the diffusion of photons out of the Sun(see Prob. 3.7), though in that case, the medium was opaque and scattering was in random directions.


For a plasma, we can express the index of refraction using its definition (Table I.1)14

together with Eq. (E.7), and writing the result in terms of the frequency, , of theincoming wave,

n ¼ 1 p

2 1=2ð5:6Þ

where p ¼ !p=ð2Þ is the plasma frequency, !p being the angular plasma frequency

(see Appendix E). Note that we require >p for n to be real, implying that a signal

with a frequency less than p cannot propagate through the plasma. Such a signal is

reflected, rather than transmitted (see Appendix E for examples.) On the other hand, if

p, then n 1 in which case there is no bending. Eqs. (5.6) and (E.6), and a

binomial expansion for small p= yield,

n ¼ 1 ne0:0063

MHz

" #2

ð5:7Þ

where ne is the electron density.

Refraction will be greatest for media in which n differs the most from 1. Therefore,

from Eq. (5.7), we can see that refraction is most important at low frequencies. Since

the lowest radio frequencies observed are seldom less than several hundred MHz, this

14As indicated in the table, the index of refraction can be written as a complex number for cases in which somefraction of the incoming light is absorbed. For the astrophysical plasmas discussed here, a complex index ofrefraction is not needed.

Figure 5.11. This simple diagram is a nice illustration of how refraction can be thought of interms of both interactions between photons and individual particles, as well as a change in thedirection of the wavefront. The arrows show the direction of propagation of the wavefront and thediagonal lines (upper left to lower right) indicate the direction of propagation of the photons. Inthis illustration, the photons themselves do not change direction when they interact with particles(consistent with scattering that is preferentially along the line of sight). The speed of light is slowerin medium 2 because there are more interactions in this denser medium. (Reproduced courtesyof Shu, F. H., The Physical Universe, An Introduction to Astronomy, University Science Books)

5.3 THE WAVEFRONT REDIRECTED – REFRACTION 173

expression shows that departures of the index of refraction from unity and therefore

changes in the angle of propagation of the signal are very small unless electron

densities exceed 107 cm3 (for example, the coronas and atmospheres of stars).

However, both a sufficient electron density as well as some specific geometry, such as

sheets or filaments, are required for refraction from ionized gas to be important in any

systematic sense. Note that Eq. (5.6) implies that n 1, (since > p is required for

propagation) so an intervening plasma can act like a diverging lens15 in front of a

background signal. If we repeated the atmospheric bending example given in the

Appendix to Chapter 2 for the ionosphere, for example, the bending of light would

be away from the normal in Figure 2.A.1.

From the point of view of astrophysical gases, refraction is mostly observed in terms

of its random effects. Even in the very low density ISM ðne 103 cm3Þ, forexample, refraction can be important for background point sources such as pulsars

or distant point-like radio-emitting quasars. As the signal travels through regions of

varying density in the ISM en route to the Earth, even a very small change in angle due

to refraction can move the signal in and out of a direct path to the detector on Earth.

This, along with other effects such as time variable scattering, causes the brightness of

the signal to fluctuate with time, an effect called interstellar scintillation. This effect, a

kind of space weather, is similar to the scintillation (Sect. 2.3.4) that we have already

seen for the Earth’s atmosphere, so the ISM acts like a cosmic atmosphere. Since the

refractive index is frequency dependent (Eq. 5.7), the ISM is also dispersive, like a

prism. That is, if a broad spectrum of wavelengths are present in the incident beam

(which is the case for pulsars and AGN) then the lower frequency light will be bent

more than the higher frequency light.

Pulsars, in particular, tend to move rapidly so, statistically, the component of their

velocity that is transverse (i.e. perpendicular to the line of sight) will be rapid as well

(of order 100 km s1). Therefore, a background pulsar will ‘illuminate’ different parts

of the ISM with time. Studying the properties of the resulting signal as a function of

both time and frequency provides important information on the properties of the ISM

and ISM turbulence. Scintillation due to refraction, for example, is variable on times

scales of days for a pulsar at a distance of 1 kpc moving at 100 km s1 and an ISM

electron density fluctuation of order a factor of 2. The implied size scale for the

perturbing cloud is then 1:5 1012 cm (Ref. [146]), or one tenth of an astronomical

unit. If dispersive properties of the ISM are considered, size scales of 1010 cm or

smaller can be studied. Estimates for the size scales of interstellar density fluctuations

that have been studied in this way range from about 107 cm to 1018 cm and the number

density of clouds follows the power law N / D11=3, where D is the cloud diameter.

Thus, analysing the perturbation in the signal from a background source is a powerful

tool for studying the richness of the structure of the ISM on size scales that would

otherwise be inaccessible to direct imaging (Prob. 5.13).

15From the definition of n, this implies that >c, but here is the phase velocity (see footnotes to Table I.1) whichcan be greater than c without violating Special Relativity since it is the group velocity that carries information.


5.4 Quantifying opacity and transparency

5.4.1 Total opacity and the optical depth

As we have seen from the previous sections, there are many possible ways in which a

photon can interact with matter. A key parameter that helped to determine whether an

interaction might occur was the mean free path, l (Eq. 3.18) already discussed in a

variety of contexts for both particle–particle collisions as well as photon–particle

collisions (e.g. Examples 3.4, 3.5, 5.2, 5.3). Although the mean free path is a

‘representative’ path length, not every photon (or particle) will travel exactly this

distance before interacting. In addition, we have discussed gases in terms of whether

they are ‘transparent’ (mean free path is greater than the size of the gas, such as the

Earth’s clear sky) or ‘opaque’ (mean free path is less than the size of the gas, such as the

Sun), as illustrated in Figure 4.2.We nowwish to quantify these terms. That is, we need

to consider what fraction of the incoming beam will be scattered out of the line of sight

or absorbed and what fraction will be transmitted through the gas. Such an approach

will allow the characterization of cases that are intermediate between being fully

opaque and fully transparent, such as a cirrus cloud in front of the Sun, a mist around a

street lamp, or a partially transparent HI cloud in the ISM. Since we wish to fold

together all processes that remove photons from a line of sight, scattering and absorp-

tion are considered together as a total opacity (or extinction which is used for dust, see

Sects. 3.5.1, 5.5).

We now introduce a unitless parameter called the optical depth, , such that an

infinitesimal change in optical depth, d , along an infinitesimal path length, dr, is

given by,

d ¼ n dr ¼ dr ¼ dr ð5:8Þ

where is the effective cross-section for the interaction (cm2), (cm2 g1) is the

mass absorption coefficient and (cm1) is the absorption coefficient, at a given

frequency. The quantities, n, and are the particle density (cm3) and mass density

(g cm3), respectively. The term, ‘absorption’, in the above context, is meant to include

all processes that remove photons from the beam (as in Figure 5.1.a and b)16. These

terms could be separated into a true absorption and a scattering portion, separately, if

desired. The negative signs in Eq. (5.8) indicate that, while the coordinate, r, increases

in the direction that the incoming beam is travelling, the optical depth is measured into

the cloud as measured from the observer (see Figure 6.2). The total optical depth is then

determined from an integration over the total line of sight through the source. Using the

version with the mass absorption coefficient,

¼ Z 0

l

dr l ð5:9Þ

16The coefficients could also include corrections for scattering into the line of sight.

5.4 QUANTIFYING OPACITY AND TRANSPARENCY 175

where we integrate from the near to the far side of the cloud and l is the line of sight

thickness of the cloud. The latter approximation is often used for situations in which the

density and mass absorption coefficient can be considered constant through the cloud.

The optical depth provides a quantitative description of how transparent or opaque a

gas is. If < 1, then the probability that the photon will be scattered or absorbed is< 1

and the cloud is optically thin or transparent (Figure 4.2 top). If > 1, then the

probability that a photon will be scattered or absorbed is high and the cloud is optically

thick (Figure 4.2 bottom). If ¼ 1, then the cloud is ‘just’ optically thick and the mean

free path is equal to the line of sight thickness of the cloud. The optical depth can be

thought of as the number of mean free paths through an object. The shape of the

function, , when plotted against frequency shows how the effective absorption

cross-section varies with frequency for a cloud of give density (Eq. 5.8). From Eq.

5.8 it is clear that the optical depth depends on (a) the type of material, (b) the

frequency of the radiation, (c) the density of material and (d) the line of sight distance.

It is possible, for example, for a small dense cloud to have the same optical depth as a

large diffuse cloud of the same material. It is also possible for two clouds of the same

size and density, but different material, to have very different optical depths. Example

5.6 further elaborates.

Example 5.6

Consider two ionized clouds of pure hydrogen, each with the same fractional ionization of99:9 per cent, density of 102 cm3 and temperature of T ¼ 104 K. Cloud 1 has a line of sightthickness of l ¼ 1 pc and Cloud 2 has l ¼ 100 pc. Compare the optical depth of Cloud 1 at awavelength of ¼ 121:567 nm to the optical depth of Cloud 2 at a wavelength of ¼ 200 nm. Assume that only scattering processes are important (see Table 5:1) and ignorescattering into the line of sight.

Cloud 1: Most particles are ionized, so electron scattering must be considered. In

addition, however, the incoming photons are exactly at the wavelength of the resonance

scattering of the Ly line (Table C.1) so we also need to consider whether Ly scattering

will be important for those few particles that are neutral (all of which we assume are in their

ground states, Sect. 3.4.6). Then, from Eqs. (5.8) and (5.9),

¼ n l ¼ T ð0:999Þ n lþ Ly ð0:001Þ n l ð5:10Þ

¼ ð6:65 1025Þ ð0:999Þ ð102Þ ð3:09 1018Þ ð5:11Þ

þ ð5:0 1014Þ ð0:001Þ ð102Þ ð3:09 1018Þ ð5:12Þ

¼ 2 108 þ 1:5 ¼ 1:5 ð5:13Þ

where we have used the Thomson scattering and Ly cross-sections from Table 5.1. This

cloud is optically thick, with an opacity dominated by Ly scattering from a small fraction

of neutral particles.


Cloud 2: In this case, the incoming photons are not at the frequencies of any resonance

lines so we need only consider Thomson scattering,

¼ ð6:65 1025Þð0:999Þð102Þð100Þð3:09 1018Þ ¼ 2 106 ð5:14Þ

This cloud, although 100 larger than the previous one, is highly optically thin.

As indicated above, the total mass absorption coefficient of Eqs. (5.8) and (5.9) must

include all processes that may be occurring. We can therefore write a general expres-

sion for that takes this into account,

¼ bb þ bf þ es þ ff þ . . . ð5:15Þ

where the subscripts indicate, bb: bound–bound processes such as line absorption and

scattering, bf: bound–free processes such as ionization, ff: free–free processes such as

free–free absorption, es: electron scattering (Thomson scattering), and whatever other

processes might be occurring17. Fortunately, not all processes are occurring in any

given environment. In fact, in many situations, a single process may dominate all

others, as suggested in Example 5.6 for the second cloud.

In the case of stars, all of the processes listed in Eq. (5.15) do have to be considered.

This is not an easy or straightforward process since some of these quantities, them-

selves, depend on density and temperature and can also be highly frequency dependent.

Bound–bound opacities, bb, for example, will be significant at frequencies corre-

sponding to the quantum transitions in atoms and zero otherwise. For the bound–free

and free–free absorption coefficients, it has been found that they can be approximated

by functional forms that depend on mass density, , and temperature, T (see Ref. [152]

for further details),

bf ¼ 4:34 1025 fbf ðZÞ ð1 þ XÞ

T3:5cm2g1 ð5:16Þ

ff ¼ 3:68 1022 gff ð1 ZÞ ð1 þ XÞ

T3:5cm2g1 ð5:17Þ

where X, Y, and Z represent the stellar composition (Sect. 3.3.1). The correction

factors, fbf and gff , are of order 0:01 ! 1 and 1, respectively18. The ‘overline’

indicates that the quantity is a Rosseland mean opacity for the process being con-

sidered, which is a statement that a weighted average over frequency has already been

carried out. For example, ff can be derived from the free–free absorption coefficient

17There could be other sources of opacity in special environments. For example, some very high energy processessuch as Compton scattering or the absorption of a high energy photon in an atomic nucleus are possibilities, orabsorption processes specific to extremely dense environments such as those that are degenerate (see Sect. 3.3.2).18The factor, fbf ¼ gbf=t, where gbf is the mean bound–free Gaunt factor for the various types of particles (of order1, see also Eq. C.9), and t is called the guillotine factor which is of order 1 to 100. The factor, gff is the free–freeGaunt factor (also of order 1) averaged over frequency. See Figure 8.2 for frequency-dependent values (Sect. 8.2).


given in Eq. (8.3), with appropriate weighting (see Sect. 8.2). Any opacity that has the

functional form/ =T3:5 is referred to as aKramers’ law.Although these functions are

approximations, they do show how the opacities depend on density and temperature.

The bound–free opacity is dominated by metals in the interior of the Sun since

hydrogen and helium are mostly ionized. However, the free–free opacity is dominated

by free electrons from hydrogen and helium because these elements are far more

numerous than the metals (Ref. [152]). Note that, as the temperature increases, the bf

opacity decreases becausemore atoms are already ionized and therefore there are fewer

bound electrons to release from the atoms. The ff opacity also decreases because, at

higher temperature, the free electrons are moving faster, on average, and have a lower

probability of absorbing an incident photon given the shorter time spent near nuclei.

For electron (Thomson) scattering in regions of complete ionization (Ref. [144]),

es ¼ 0:200 ð1 þ XÞ cm2g1 ð5:18Þ

(Prob. 5.16), where a mean over frequency is not necessary since Thomson scattering is

frequency independent (Appendix D.1.1).

In stars in which there are few heavy metals, the most important processes are free–

free absorption and electron scattering. In stars with significant metallicity and in the

cooler outer layers of stars in which there are more bound electrons, bf will dominate.

The most important source of opacity in the outer layers of the Sun, for example, is

bound–free absorption from the H ion. It is clear that the opacity of stars is a very

sensitive function of density, temperature, and composition, and can vary strongly

between the deep interior and the surface. See Ref. [70] for opacity values that are

consistent with the Solar interior model of Table G.10.

Determining the opacities in stars and elsewhere is an extremely important pursuit in

astrophysics and many astronomers have spent their careers calculating and improving

upon the published values. Without a knowledge of opacities, we could not understand

how the signal is affected as it travels through matter, be it stellar, interstellar, atmo-

spheric, or any other material (this is pursued in greater detail in the next chapter).

Without a knowledge of opacities in stellar interiors, moreover, we could not relate the

luminosities that are observed at the surfaces of stars to the sources of energy

generation at their cores, nor could other stellar parameters be accurately determined

as a function of radius, such as densities, temperatures, ionization states, and others. In

fact, the very structure of a star is intimately connected with its opacity and the

variation of opacity with radius. As we will see in the next section, the opacity can

have important consequences for stellar dynamics as well.

5.4.2 Dynamics of opacity – pulsation and stellar winds

If an incident beam passes through a gas with little probability of interaction ð 1Þthen that beam will have little or no dynamical effect on the gas. However, if the

probability of absorption is very high ð 1Þ then, as indicated in Sects. 1.5 and


5.2.1, the photons exert a pressure against the gas, as if hitting a wall. Thus, anywhere

that the opacity is sufficiently high, dynamical effects can become significant. The

following are two important examples of the dynamics of opacity.

The first occurs in certain types of variable stars (stars that vary in brightness) that do

so because they are radially expanding and contracting. Cepheid variables are exam-

ples of this, named after the proto-type star, Cephei. These stars are essential tools inobtaining distances in the Universe because they are bright enough to be seen over

large distances and also because their pulsation periods (which are easily measured and

vary from about 1 to 100 days) are related in a known way to their intrinsic luminos-

ities. If the apparent magnitude of a Cepheid is measured and the intrinsic luminosity is

inferred from the pulsation period, then the distance can be found using Eq. (1.30).

Objects like this, whose intrinsic luminosities are known, are called standard candles.

Since Cepheids are very bright and can be seen in galaxies other than our own Milky

Way, they are used to measure distances out to about 25 Mpc.

If a small perturbation (compression) occurs in a layer within a star, both the

density and temperature will increase slightly. From Kramers’ laws of Eqs. (5.16)

and (5.17), the temperature has a greater effect than the density so the mass

absorption coefficient normally decreases. Since ¼ (Eq. 5.9), a decrease in

in the perturbed layer offsets the increase in so that there is little change in the

optical depth. Thus, stars tend to be dynamically stable to such perturbations.

However, there are certain regions within stars that are called partial ionization

zones. These are at depths corresponding to temperatures at which hydrogen and

helium (the latter corresponds to a deeper layer) become ionized. In these layers, if

there is a small compression, instead of the temperature increasing as it did before,

energy goes into ionization instead, which is a change of phase. The effect is similar

to one in which energy is continually added to ice, raising its temperature to the

melting point at which time the energy goes into the ice-to-water phase change rather

than an increase in temperature. Since the temperature rise in the stellar ionization

zones is now very small, the opacity, described by Kramers’ Laws, increaseswith the

increasing density of the perturbed layer. The optical depth then also increases.

Photons from the deeper interior then exert a greater pressure against the perturbed,

higher opacity layer, causing an expansion. The opposite occurs during expansion as

electrons recombine, releasing energy, increasing the temperature and therefore

decreasing the optical depth again. Thus, during compression the partial ionization

zone absorbs heat and during expansion the layer releases heat, acting like a heat

engine. During expansion, the layer may overshoot, that is, the velocity given to the

layer by radiation pressure perturbs it from gravitational (or hydrostatic) equilibrium

and therefore it will fall back again gravitationally, causing the cycle to repeat. This

process is called the kappa mechanism after the mass absorption coefficient, which

acts like a valve, within the star.

Not every star experiences such pulsation. Stars that are too hot have ionization

zones too close to the surface in regions in which the density is too low for the kappa

mechanism to be effective. Stars that are too cool have deep convection zones which

disrupt the kappa mechanism. Thus, only a certain region of the HR diagram


(Figure 1.14) is amenable to this pulsation mechanism. This region is referred to as the

instability strip.19 Stellar pulsation can result in mass loss and, if the outer layers of the

pulsating star are extended and of low density, such as in a red giant star, then mass loss

is more likely. Opacity-driven dynamical effects, such as the kappa mechanism,

therefore play a crucial role in the evolution of stars (see Sect. 3.3.2).

The second important example involves stars, or other objects, of very high lumin-

osity. Most stars, like our Sun, are stable because their internal pressure provides

support to balance their inwardly pulling gravity. The internal pressure is the sum of all

internal pressures (see Eq. 3.16) which, for lowmass stars like our Sun, is dominated by

particle pressure given by the perfect gas law (Eq. 3.11). However, in stars that aremore

massive, hotter, and more luminous, as described in Sect. 4.1.5, the radiation pressure

(Eq. 4.15) dominates. In very highmass stars, the radiation pressure is so strong that the

star’s gravity is insufficient to retain the outer gaseous layers and they are blown off in

stellar winds (Figure 5.10). This, in fact, puts a theoretical upper limit on the masses of

stars. In Sect. 3.3.2, we noted that a lower limit to the mass of a star is set by the

inability of low mass stars to ignite nuclear burning in the core. Now we have an

argument for an upper limit to the mass of a star. Although other processes can also

contribute to stellar mass loss (such as magnetic fields) the simple argument of

radiation pressure leads to a theoretical upper limit that is very close to the observed

upper limit of stellar masses (see Example 5.7). The highest luminosity that any object

can have before radiation pressure starts to ‘blow the object apart’ is called the

Eddington Luminosity or Eddington Limit. Our Sun’s luminosity is ‘sub-Eddington’

because it is not shining at the maximum luminosity possible for an object of its mass.

However, we should not see any objects that are ‘super-Eddington’, because they

would not be stable. It has been suggested that some super-Eddington examples exist,

but these may be related to special circumstances such as short time scales, inhomo-

geneous gas distributions, or unusual geometries.

Example 5.7

Estimate the highest stellar mass possible before radiation pressure starts to cause mass loss.

Consider a star with an outer layer of thickness equivalent to a photon’s mean free path,l (taken to be small in comparison to the star’s radius). Then a typical photon that passes

through this region will be absorbed and ¼ l ¼ 1. The pressure in such a layer is

directed outwards and we can then use Eq. (1.23) with ¼ 0, to find the radiation pressure

on this outer layer,

Prad ¼f

c¼ L

4 r2 cð5:19Þ

19Note that there are a variety of different kinds of variable stars and also a variety of pulsation mechanisms. Theinstability strip shown in Figure 1.14 is applicable only to Galactic Cepheids [Ref. 166a].


where f is the star’s flux, L is its luminosity, r is its radius, and c is the speed of light. The

force per unit area due to gravity acting inwards on this layer is,

Pgrav ¼F

4 r2¼ 1

4 r2GM m

r2¼ GMðVÞ

4 r2 r2¼ GM 4 r

2l

4 r2 r2ð5:20Þ

where m is the mass of the layer, is its density and V ¼ 4r2l is its volume. For balance,

we equate Eq. (5.19) to Eq. (5.20) and replace l with 1= to give,

LEd ¼4 cGM

ð5:21Þ

where LEd is the Eddington Luminosity. That is, if L > LEd, then the star will start to lose

mass through radiation pressure. It is therefore is unlikely that stars with luminosities

greater than LEd exist. For the very high temperature, low density outer layers of the most

luminous, massive stars, the dominant source of opacity should be electron scattering.

Therefore, using Eq. (5.18) with X ¼ 0:707 (Eq. 3.4), expressing the mass in units of Mand the luminosity in L,

LEd ¼ 3:8 104M

ML ð5:22Þ

It is now possible to compare this luminosity to the highest luminosities that are actually

observed for stars. The most massive stars known (Ref. [103]) have M 136 M, forwhich LEd ¼ 5:2 106 L (Eq. 5.22). These stars are observed to have Mbol 11:9which, using Eq. (1.31), corresponds to an observed luminosity of Lobs ¼ 4:5 106 L.Thus the observed upper limit is in good agreement with the Eddington value.

The Eddington limit applies to objects other than stars as well. For example, the

Eddington luminosity argument is often applied to active galactic nuclei (AGN) that

exist at the centres of some galaxies (e.g. Figures 1.4, 5.5). AGN activity is believed to

be powered by a supermassive black hole, that is, black holes in the range, 105–1010

M. Although no light can escape from the black hole itself, the material that is

surrounding it, either accreting onto the black hole or rapidly orbiting it, is in a region in

which the gravitational energy is extremely high. This material is heated to very high

temperatures, producing radiation and its resulting pressure. Just as for stars, the

radiation pressure must not exceed that produced by the gravity of the black hole itself

or else the material will be blown away. It is believed that most AGN are radiating at or

close to their Eddington limits (within a factor of 3, Ref. [90]). Thus, Eq. (5.22)

provides a way of relating an observable AGN luminosity to the mass of an unobser-

vable black hole. Typical values of luminosity for powerful AGNs are in the range,

1045 ! 1047 erg s1, corresponding to black hole masses of 107 ! 109 M at the

centres of galaxies.


5.5 The opacity of dust – extinction

The interaction of light with dust grains is complex since dust grains have irregular

shapes, come in a variety of compositions with individual grains also being of non-

uniform composition, span a range of sizes, and some fraction of them may be aligned

with themagnetic field (see Sect. 3.5). As a result, details of the interactionmay involve

scattering, absorption, diffraction, and polarization.

A convenient starting point is to assume that the grains are spherical, span a range of

sizes in comparison to the incident light, and have uniform composition (though the

composition may differ for different grains). An extensively used theory for dust is

called Mie Theory which folds the absorption and scattering characteristics of a

spherical grain into a complex index of refraction, as described more fully in Appendix

D.3. The result is a determination of how efficiently light interacts with the grain as a

function of wavelength and grain size. This is measured by the extinction efficiency

factor, Qext, of a dust grain (the sum of scattering and absorption components) as a

function of x ¼ 2a= where Qext is the ratio of the effective to the geometric

extinction cross-section of a grain, a is the grain radius and is the wavelength of

the incident light. A plot ofQext, including its absorption and scattering parts, is shown

in Figure D.5 for one specific grain composition. The strongest interaction occurs when

the wavelength of the light is of order the grain size.

Using Eqs. (5.9), (5.8), and (D.30), the optical depth of a set of grains, assuming that

there is no change in properties as a function of line-of-sight distance, is,

¼ nd l ¼ nd Qext a2 l ð5:23Þ

where nd is the number of grains per unit volume, is the effective grain cross-section(equivalent to Cext in Appendix D.3) and l is the line-of-sight depth of the region.

Once Qext is determined for a variety of different grains, it is then possible to

calculate a theoretical extinction curve for various admixtures of grain sizes and

compositions (see Example 5.8 for a simplified example, and Prob. 5.18) and

compare the result with the observed extinction curve shown in Figure 3.21. The

admixture which produces the closest fit is more likely to represent the dust that is

actually present.

Example 5.8

Assuming that all grains are identical and possess the properties illustrated by Figure D.5,describe how Figure D.5 can be related to the observed extinction curve shown in Figure 3:21.

From Eqs. (1.13) and (3.29), the absorption in some waveband, , can be written,

A ¼ 2:5 logI

I0

ð5:24Þ


where I is the specific intensity of the object undergoing extinction and I0 is the specific

intensity in the absence of extinction. Anticipating a result from Sect. 6.2 for the transmis-

sion of light through a medium that absorbs, but does not emit optical light (Eq. 6.9),

I

I0¼ e ð5:25Þ

Combining the above two equations,

A ¼ 1:086 ð5:26Þ

This result shows that an optical depth of one is approximately equivalent to an extinction

of one magnitude20 Using Eqn. (5.23),

A ¼ 1:086 nd Qext a2 l ð5:27Þ

From this equation, Eq. (3.33), and our assumption that there is no change in dust properties

along the line of sight,

EV

EBV

¼ Qext QextV

QextB QextV

ð5:28Þ

The only wavelength-dependent quantity of the right hand side of this equation is Qext.

Since we have taken the grain size to be constant, the abscissas of the Qext plot

(Figure D.5) and the extinction curve (Figure 3.21) both represent 1=. Therefore, if allof our assumptions are correct, then the shape of the curve of Qext (except for some

scaling) should be the same as the shape of the extinction curve. A visual comparison

indeed shows that there is some similarity. However, there are also significant differences,

leading to the conclusion that all dust grains (if described by figure D.5) are not identical.

Problems

5.1 What would be the scattering cross-section of a photon from a free proton (see

Eq. D.2)? In an ionized gas, what fraction of scatterings will be due to free protons in

comparison to free electrons?

5.2 (a) From the relationships in Appendix D.1, show that the Einstein Aj;i coefficient

between energy levels, j and i can be written,

Aj;i ¼0:667

2i; j

gi

gjfi; j ð5:29Þ

20 This would be true for any medium (dust or gas) that does not, itself, produce emission at the same wavelengthas the incident light.

5.5 THE OPACITY OF DUST – EXTINCTION 183

where i; j is the wavelength of the transition, gj, gi, are the statistical weights of the upper

and lower states, respectively, and fi; j is the absorption oscillator strength. Verify that this

relationship gives the Einstein Aj;i coefficients of the Ly g and H g lines given in Table C.1.

(b) Show that the cross-section given by Eq. (D.12) can be written,

s ¼21;2

2

g2

g1

ð5:30Þ

for the centre of the Ly line in the absence of any other line broadening mechanisms,

where 1;2 is the wavelength of the line centre. Verify that this result gives the natural

bound–bound cross-section given in Table 5.1.

5.3 For a scattering process in which a Ly g photon is absorbed by hydrogen followed by

the emission of an H and then Ly photon, show that the absorbed energy is equal to the

total emitted energy.

5.4 Consider resonance scattering from bound electrons in the hydrogen atom (Appen-

dices D.1.2, D.1.3, and D.1.4) in which the line widths are given by the natural line width21.

(a) Which of the spectral lines in Table C.1 has the widest frequency response to photon

scattering?

(b) Under typical interstellar conditions, almost all neutral hydrogen atoms are in their

ground states (see Sects. 3.4.5, 3.4.6). Assuming that there are an equal number of photons

at all frequencies impinging upon a hydrogen atom in the ISM, which transition is the most

probable and why?

5.5 Consider Rayleigh scattering from bound electrons (Sect. 5.1.1.3).

(a) In the Rayleigh scattering limit for any atom or molecule, how much greater (i.e. by

what factor) will the scattering cross-section be for blue light in comparison to red light

(Table G.5)?

(b) Calculate the total Rayleigh scattering cross-section, R, for the hydrogen atom at

the twowavelengths adopted in part (a) using data from Table C.1 and Eq. (D.24). Consider

only the lines for which the Rayleigh limit is applicable. Determine the ratio of cross-

sections for the two wavelengths and comment as to why there might be a difference with

the results of part (a).

5.6 In Example 5.3, the mean free path of a photon in the Earth’s atmospherewas found to

be greater than 80 km. Above this altitude, the atmosphere is largely ionized (the iono-

sphere). From the information in Sect. 2.3, calculate the mean free path of a photon to

Thomson scattering in the ionosphere. Is the ionosphere more or less transparent to optical

photons than the atmosphere at sea level?

21Note that the natural line width is much narrower than observed in nature (see Sect. 9.3).


5.7 (a)What is the mean value of the increase in wavelength,, for Compton scattering

in an isotropic radiation field?

(b) Show that, after N Compton scatterings of equal angle, , each time, the final photon

energy, EN , can be written in terms of the initial photon energy, Ei, via,

EN ¼ Ei

1 þ N Ei

me c2ð1 cos Þ

ð5:31Þ

(c) If a photon of energy, 5.1 MeV is repeatedly Compton scattered at an angle of 60,how many scatterings are required to reduce its energy to 0.51 MeV?

5.8 What is the mean free path of a 0.51 MeV photon to Compton scattering through (a)

the intergalactic medium of mean density, n ¼ 104 cm3 and (b) through a Wolf–Rayet

star of mass, 20M, and radius, 20 R? Comment on how easily this gamma-ray photon

can escape from the star and how easily it can travel through intergalactic space in the

absence of other interactions.

5.9 Determine the Stromgren radius of the resulting H II region if the Sun were to enter an

H I cloud of the same density as the H II region described in Example 5.4 i.e. let ne after

ionization ¼ nH before. Assume that the H II region temperature is also the same. Based on

the result, qualify the statement: ‘only O and B stars create H II regions’. [Hint. Either

BðTÞ or BðTÞ may be used. Consider whether either the Rayleigh–Jeans Law or Wien’s

Law might be used to simplify the integration.]

5.10 Suppose a spherical, pure H I primordial galaxy of uniform density is present in

the dark ages (Sect. 5.2.2) before any stars or AGN exist. The galaxy has a mass and

radius of MG ¼ 1010 M and RG ¼ 10 kpc, respectively. If an AGN of ionizing

luminosity, Li ¼ 1046 erg s1 suddenly ‘turns on’ at the centre of this galaxy, would

the entire galaxy become ionized and, if so, would photons leak out into the inter-

galactic medium? If not, what fraction of the galaxy would be ionized? For simplicity,

assume that all ionizing photons from the AGN are exactly at the energy required to

ionize hydrogen.

5.11 Consider the development given in Appendix E and write the plasma frequency

equation (Eq. E.6) for ions of charge, Ze, instead of electrons. Determine the ratio of the

plasma frequency oscillation period of ions compared to electrons for a pure hydrogen gas.

5.12 (a) Compute the electron plasma frequency, p, for a typical maximum density of

ne ¼ 106 cm3 in the Earth’s ionosphere (Sect. 2.3).

(b) Look up the FM, AM, and shortwave radio band ranges from some convenient

reference. Indicate which frequencies in these bands, emitted by a ground-based transmit-

ter, could reflect off of the ionosphere and therefore be detected by a radio receiver that is

over the physical horizon.

5.5 THE OPACITY OF DUST – EXTINCTION 185

(c) Referring to Figure E.2 of a Type III radio burst, estimate the maximum and

minimum electron densities in the plasma between the Sun and the Earth encountered

by this Solar flare.

(d) Indicate over what range of frequencies the radio burst of part (c) could have been

detected from the Earth’s surface. How much of the emission shown in Figure E.2 could

have been detected from the Earth’s surface?

5.13 What would be the angular size (arcsec) subtended by the smallest interstellar cloud

described in Sect. 5.3 if it were at the fairly nearby distance of 50 pc? Determine how large

a radio telescope (let ¼ 20 cm) would be required to image such a cloud directly.

Comment on the technique of using interstellar scintillation for probing these size scales

in comparison to direct imaging.

5.14 Compute optical depths (see Table 5.1) for the following cases and indicate whether

the cloud or medium is optically thick or optically thin:

(a) 5.1 MeV photons travelling 100 Mpc through intergalactic space of density of

ne ¼ 104 cm3.

(b) An incident beam containing 13.6 eV photons impinging on an ISM H I cloud of

density, nH ¼ 1 cm3 and thickness, l ¼ 1 pc.

(c) Incident photons at 500 nm in an upper layer of a star that is 100 km thick with a

density, ¼ 108 g cm3 and a total mass absorption coefficient of 500 nm ¼0:83 cm2 g1.

5.15 Consult Table G.11 and explain why the Sun’s limb appears sharp to the unaided eye

(assuming no glare as would be the case if the Sun is viewed behind a thin, partially

transparent cloud).

5.16 Using the Thomson scattering cross-section, T ¼ es, derive Eq. (5.18). Eqs. (5.8)

and (3.14) will be useful.

5.17 From the information given in Table G.10 or Figure G.2, estimate bf , ff , and es at

positions, (a) R ¼ 0:1R, (b) R ¼ 0:5R, and (c) R ¼ 0:9R, within the Sun. Adopt

fbf ¼ 0:1 and gff ¼ 1. Comment on the relative importance of these three processes

(bound–free absorption, free–free absorption, and electron scattering) as a function of

depth.

5.18 Estimate EV

EBVfor a wavelength, 210 nm, assuming that all dust grains are identical

with radii of a ¼ 0:1 mm and have the properties shown in Figure D.5 (see Example 5.8).

Compare the result to the observed value shown in Figure 3.21. Estimate how much this

would change if all grains had radii a factor of 2 larger and comment on the results.


6The Signal Transferred

Behold! Only 34 circles are required to explain the entire structure of the universe and the dance of

the planets!

–Nicolaus Copernicus in Commentariolus (Ref. [64]).

6.1 Types of energy transfer

The fact that we see light in the Universe means that the energy contained in that light –

a tiny fraction of what was actually generated – has been transported from its point of

origin to our eyes or our telescopes. With no intervening matter, the energy transferred

is just that contained in photons diminished by distance (e.g. Eq. 1.9). With intervening

matter, as we have seen, photon–particle or particle–particle interactions1 can transfer

or ‘share’ energy. We have so far considered how such interactions might divert a

photon out of (or possibly into) the line of sight. We now wish to consider the net

transfer of energy along a path as a beam of light moves through matter. This will allow

us to relate the interactions that are occurring at a microscopic level to the light that we

actually observe. In the process, we can obtain information about the material within

which energy transport is occurring.

Energy can exist in several forms. Kinetic energy, which is the energy of motion, is

called ‘heat’ when applied to random particle motions as discussed in Sect. 3.4.1. The

energy of a photon and the energy equivalent of matter are given by E ¼ h n and

E ¼ m c2, respectively (Table I.1). Example 3.1 used the latter expression to explain

the luminosity of the Sun. Potential energy exists only with reference to a ‘field’. In a

gravitational field, for example, a higher object ‘contains’ more gravitational potential

energy than a lower one. If let go, the potential energy of the object will be converted

into the kinetic energy of motion. In an electric field, two like charges near each other


1As mentioned in Sect. 5.1.2.2, photon–photon collisions are also possible, but they are a less commonmechanism for transferring energy in astrophysics.

contain more electrical potential energy than two like charges separated by a large

distance. Again, if let go, the potential energy will be converted to kinetic energy as the

like charges repel each other. An emitted photon resulting from the transition of an

electron from a higher quantum state to a lower one can be thought of as the conversion

of electrical potential energy into radiative energy. In discussing energy transport,

therefore, it is useful to keep in mind the kinds of energy exchanges that can occur.

The transport of energy along a path through matter can occur in four ways:

(a) thermal conduction, (b) convection, (c) radiation, and (d) electrical conduction.

The first three are routinely cited as heat transport mechanisms in non-astronomical

objects. Thermal conduction involves collisions between particles, and therefore the

sharing of particle kinetic energy, with no net bulk motion of the material. If one side of

a material is hot and the other cold, eventually collisions between adjacent particles

will cool the hot side and warm the cool side. Convection involves the physical motion

of a fluid (liquid or gas) and only exists in the presence of a gravitational field. In a gas,

hotter pockets (‘cells’) move upwards into cooler regions via buoyancy and start a

circulation or ‘boiling’ motion2. Energy is thus transported by the physical motion of

hot gas upwards. Radiation is taken to mean the transport of energy via emission and

eventual absorption again of photons, as we considered at length in Chapter 5.

Electrical conduction refers to the motion of electrons through a medium in which

the nuclei are fixed. It is rarer in astrophysics but it is an important mechanism for

transporting energy in white dwarfs. Electrical currents can also exist in some regions

in which there are strong magnetic fields, such as regions near Sunspots.

In our own Sun (see Figure 3.15), radiative transfer and convection are the important

mechanisms for transporting energy from the core to the surface. In principle, thermal

conduction can also occur, but its efficiency is low in comparison to the other

mechanisms. Electrical conduction will not be important until the Sun evolves off

the main sequence and the Solar core becomes dense enough to be considered a white

dwarf precursor (Sect. 3.3.2). Presently, radiative diffusion dominates out to about

70 per cent of the Solar radius while, in the outer 30 per cent, energy is primarily

transferred via convection which dominates when the temperature gradient is high3.

The granulation seen at the surface of the Sun (Figure 6.1) is a beautiful illustration of

convective cells. In other stars, the same energy transfer mechanisms are occurring

although the interior locations and relative importance of radiation and convection may

differ. Stars of lower mass than the Sun have deeper convective zones, for example,

whereas high mass stars may have no convection in the outer layers, but instead have

convective cores.

Radiative transfer is arguably the most important energy transport mechanism in

astrophysics. It involves the kinds of absorptions, scatterings, and re-emissions of

2Convection is a complex process that does not lend itself well to a detailed mathematical description. Anapproach that considers how far a particular cell will rise before its temperature matches its surroundings, and thatcalculates the resulting transfer of energy is called Mixing Length Theory. See Ref. [33] for more details.3More specifically, convection occurs when d lnðPÞ=d lnðTÞ < g=ðg 1Þ, where P is the pressure, T is thetemperature, and g CP=CT is the ratio of specific heats, where CP and CT are the amount of heat required toraise the temperature of 1 g of material 1 K held at constant pressure, and constant temperature, respectively.

188 CH6 THE SIGNAL TRANSFERRED

photons that have been described in some detail in Chapter 5 and earlier with respect to

photons traversing the interior of the Sun (Sect. 3.4.4). Beyond its importance in stars,

this is also how radiation from any background source transfers through intervening

material that may lie along its path. This includes the signal from a quasar in the distant

Universe, light from an interstellar cloud, or starlight passing through the Earth’s

atmosphere. Moreover, the intervening material itself may not only be a source of

opacity, but also a source of radiation. In the next section, then, we focus on radiative

transfer which takes into account the propagation of a signal through absorbing,

scattering and emitting matter.

6.2 The equation of transfer

Let us consider an incoming signal of specific intensity, In, passing through a cloud. By

‘cloud’, we mean any gaseous region containing matter, including sections of larger

Figure 6.1. This granulation pattern illustrates convection in the outer layers of the Sun. Brighterregions are hotter and darker regions are cooler. This section is 91 000 km 91 000 km in size andthe image was taken at a wavelength of l403:6 nm using the Vacuum Tower Telescope on Tenerifein the Canary Islands. (Reproduced by permission of W. Schmidt, Kiepenheuer-Institut furSonnenphysik) Inset: The Earth, to scale, as taken from the Galileo Spacecraft. (Reproduced bypermission of NASA/Goddard Space Flight Centre)

6.2 THE EQUATION OF TRANSFER 189

clouds, layers in the interiors of stars or their atmospheres, or whatever other gaseous

region is of interest. The cloud has some total thickness, l, and the direction, r (see

Figure 6.2) is in the same direction as the incoming beam. As the beam passes through a

small thickness of width, dr, its specific intensity will change from In to In þ dIn, where,

dIn ¼ dIn loss þ dIn gain ð6:1Þ

The first term includes all scattering and absorption processes that remove photons

from the line of sight at the frequency being observed (from now on, referred to

collectively as ‘absorptions’). The second term contains all processes that add photons

to the beam, such as all emission processes from the matter itself at the frequency being

observed4. Note that dIn could be positive or negative. Using the absorption coefficient,

an (cm1) introduced in Eq. (5.8), and introducing the emission coefficient,

jn ðerg s1 cm3 Hz1 sr1Þ, Eq. (6.1) can be written,

dIn ¼ an In dr þ jn dr ð6:2Þ

Recall that the absorption coefficient represents the mean number of absorptions

per centimetre through a material that would be experienced by photons (the inverse,

1=an, being a photon’s mean free path). Therefore, if there are more incident

photons, there will be more absorptions, which is why the loss term is a function of

In. The emission coefficient, on the other hand, is a quantity that represents all

(a)

(b)

cloud

In0

In

In

In + dIn

dr

dtn

r

tn

tn

Figure 6.2. (a) Illustration of the change in intensity of a beam of light as it passes throughmaterial of infinitesimal thickness, dr, or optical thickness, dtn. The outgoing intensity may behigher or lower than the incident intensity. (b) As in (a), but showing the background incidentbeam and the total thickness, l, or total optical thickness, tn, of the cloud. The development ofSect. 6.2 does not require the beam to be perpendicular to the cloud, as shown here.

4We ignore scatterings into the line of sight from other angles in this development.


radiation-generating mechanisms per unit volume of material and it does not depend on

the strength of the incident beam5.

Rearranging Eq. (6.2), using Eq. (5.8) and recalling that r increases in the direction

of the incident beam while tn increases into the cloud from the point of the view of the

observer, we arrive at two forms of the Equation of Radiative Transfer,

dIn

dr¼ an In þ jn ðForm 1Þ ð6:3Þ

dIn

dtn¼ In

jnan

¼ In Sn ðForm 2Þ ð6:4Þ

where

Sn jn=an ð6:5Þ

is called the Source Function and has units of specific intensity

ðerg s1 cm2 Hz1 sr1Þ. The Source Function is a convenient way of folding in all

the absorptive and emissive properties of the cloud into one term.

The solution of the equation of radiative transfer links the observed intensity, In, to

the properties of the matter, since tn contains information on the type and density

of the intervening material through Eq. (5.8). This link between the observed light and

the physical properties of matter is at the heart and soul of astrophysics and provides

the key to deciphering the emission that is seen. We will now lay the groundwork for

this process by showing the solution for some specific cases. Since Eqs. (6.3) and (6.4)

are equivalent, we will adopt whichever form is most convenient. It is also helpful to

remember that, for a single physical situation, the Equation of Transfer could be written

many times, once for every frequency for which data are available (although fre-

quency-averaging is sometimes carried out as described in Sect. 5.4.1). For example,

matter that ‘absorbs only’ at one frequency could ‘emit only’ at a different frequency

(e.g. dust at optical and infrared wavelengths, respectively, see the Sombrero Galaxy

shown in Figure 3.22).

6.3 Solutions to the equation of transfer

6.3.1 Case A: no cloud

If there is no intervening cloud, then there is no absorption or emission, i.e.

an ¼ jn ¼ 0. Using Eq. (6.3),

dIn

dr¼ 0 ) In ¼ constant ð6:6Þ

5An exception is stimulated emission which is a process by which an incoming photon causes an atom to de-excitewithout the absorption of the incident photon. This is an emission process but, since it depends on the strength ofthe background radiation, if it is occurring, it can be incorporated into the loss term as a correction factor. It istypically important only in very strong radiation fields.

6.3 SOLUTIONS TO THE EQUATION OF TRANSFER 191

Thus we come to the same conclusion as first presented in Sect. 1.3, that is, the

specific intensity is constant and independent of distance in the absence of intervening

matter.

6.3.2 Case B: absorbing, but not emitting cloud

If the cloud absorbs background radiation at some frequency, but does not emit at the

same frequency then (Eq. 6.4) jn ¼ Sn ¼ 0 and,

dIn

dtn¼ In ð6:7Þ

Integrating from the front to the rear of the cloud,

Z In0

In

dIn

In¼Z tn

0

dtn ð6:8Þ

where In0 is the initial specific intensity when the beam first enters the cloud (the

background intensity), In is the emergent intensity on the near side of the cloud, and tnis the total optical depth of the cloud along the line of sight being considered. The

solution is,

In ¼ In0 etn ð6:9Þ

We see that the intensity diminishes exponentially with optical depth.

This equation was used to describe the extinction of optical light when travelling

through a dusty medium (Example 5.8). It is also relevant to the opacity of the Earth’s

atmosphere which is an absorbing, but not emitting medium at optical wavelengths6.

Thus, Eq. (6.9) allows us to quantify the atmospheric reddening that was discussed in

Sects. 2.3.5 and 5.1.1.3 (see Prob. 6.1).

6.3.3 Case C: emitting, but not absorbing cloud

Using Eq. (6.3) with an ¼ 0, we find,

In ¼ In0 þZ l

0

jn dr ð6:10Þ

Unless we are modelling the cloud in some detail, in many cases an assumption that

the emissive properties of the cloud do not change along a line of sight (i.e.

6Since the atmosphere does show some optical emission lines, this development applies to frequencies that are offthe line frequencies.


jn independent of r) is adequate for estimating the needed physical parameters of the

cloud. For such cases, Eq. (6.10) can be written,

In ¼ In0 þ jn l ð6:11Þ

In either case, an expression for jn is required in order to relate the emission to the

source parameters. Expressions for jn for several emission processes are provided in

Chapter 8.

6.3.4 Case D: cloud in thermodynamic equilibrium (TE)

In thermodynamic equilibrium (Sect. 3.4.4), the radiation temperature is constant and

equal to the kinetic temperature, T, everywhere in the cloud. The specific intensity, In,

is therefore given by the Planck function, BnðTÞ (see Sect. 4.1) at every point in the

cloud. Thus, there is no intensity gradient (dIn=dr ¼ 0, In ¼ constant) and (Eq. 6.3)

becomes,

0 ¼ an BnðTÞ þ jn ) BnðTÞ ¼jnan

ð6:12Þ

Eq. (6.12) is known as Kirchoff’s Law and is another way of stating that emissions and

absorptions are in balance in such a cloud7. By the definition of the Source Function

(Eq. 6.5), for a cloud in TE we therefore have,

In ¼ Sn ¼ BnðTÞ ð6:13Þ

If such a cloud existed in reality, a photon that impinged on it would simply be

absorbed and ‘thermalized’ to the temperature of the cloud. However, as indicated in

Sect. 3.4.4, a cloud in true thermodynamic equilibrium is difficult to find. The best

example is the cosmic background radiation.

6.3.5 Case E: emitting and absorbing cloud

When the cloud is both emitting and absorbing, the general solution for a case in

which the emission and absorption properties do not change along a line of sight8 is

(Eq. 6.4),

In ¼ In0 etn þ Sn ð1 etnÞ ð6:14Þ

7More accurately,jnan¼ nn

2 BnðTÞ, where nn ( 1) is the index of refraction.8See Ref. [143] for the derivation of this result and also for the case in which the properties change with line-of-sight distance.

6.3 SOLUTIONS TO THE EQUATION OF TRANSFER 193

This expression allows us to see the various contributions to the observed specific

intensity. The first term on the RHS is identical to Eq. (6.9) and indicates that a

background signal will be attenuated exponentially by a foreground cloud of optical

depth, tn. The second term includes two contributions. Sn is the added specific intensity

from the cloud itself, and Sn etn accounts for the cloud’s absorption of its own

emission.

Let us now consider the two extremes of optical depth: opaque ðtn 1Þ and

transparent ðtn 1Þ. Then Eq. (6.14) reduces to,

In ¼ Sn ðtn 1Þ ð6:15Þ

In ¼ In0 ð1 tnÞ þ Sntn ðtn 1Þ ð6:16Þ

¼ In0 ð1 tnÞ þ jn l ð6:17Þ

where we have used an exponential expansion (Eq. A.3) for the case, tn 1. In

Eq. (6.17), l is the line-of-sight distance through the cloud and we have used the

definition of the Source Function (Eq. 6.5) and the equations relating optical depth to

the absorption coefficient (Eqs. 5.8, 5.9), assuming there is no variation in these

quantities through the cloud.

6.3.6 Case F: emitting and absorbing cloud in LTE

In LTE, as described in Sect. 3.4.4, the temperature of a pocket of gas is approximately

constant over a mean free photon path (see Example 3.5 for the interior of the Sun) and

therefore, over this scale, the emissions and absorptions must be in balance. We

therefore associate the Source Function with the Planck function,

Sn jnan

¼ BnðTÞ ð6:18Þ

Since there is a net flux of radiation through the cloud, however, In is not constant so

there is a non-zero intensity gradient. Then following Eq. (6.14),

In ¼ In0 etn þ BnðTÞ ð1 etnÞ ð6:19Þ

The above equation is, arguably, the most important equation in radiation astrophysics

and can be applied to any thermal cloud (i.e. one whose emission in some way depends

on its temperature), provided the gas is in LTE. As indicated in Sect. 3.4.4, moreover,

even if a cloud is not in LTE, sometimes an assumption of LTE is a useful starting point

and can provide initial, approximate parameters for the cloud. We therefore explore

some of the implications of the LTE solution in the next section.


6.4 Implications of the LTE solution

6.4.1 Implications for temperature

Eq. (6.19) leads to some important conclusions in specific circumstances. Following

the development in Sect. 6.3.5, we can write down the result for the optically thick and

optically thin extremes,

In ¼ BnðTÞ ðtn 1Þ ð6:20Þ

In ¼ In0 ð1 tnÞ þ BnðTÞ tn ðtn 1Þ ð6:21Þ

¼ In0 ð1 tnÞ þ jn l ð6:22Þ

Eq. (6.20) is a confirmation that an optically thick cloud in LTE emits as a black

body. A background source, if present, does not contribute to the observed specific

intensity (although it might contribute to heating the cloud) since the foreground cloud

completely blocks its light. Stars are good examples of this case (Sect. 4.1). The only

information obtainable from an optically thick source in LTE is the temperature since

the Planck curve is a function of temperature only. We can see into such a source only

to a depth of the mean free path which is less than the depth of the source itself (see

Figure 4.2). We have no way of obtaining any information from the far side of an

optically thick cloud and therefore no way of obtaining other information about it, such

as its density.

On the other hand, Eqs. (6.21) and (6.22) show that, for an optically thin source, the

observed intensity depends on the intensity of the background source, if present, since

the background source can be seen through the cloud. Moreover, Eq. (6.21) indicates

that, for an optically thin source, the observed intensity depends on both the tempera-

ture and density of the foreground cloud because BnðTÞ is a function of temperature and

tn is a function of density via Eq. (5.8) or (5.9). The signal that we detect has probed

every depth through the cloud in such a case. The emission coefficient of Eq. (6.22) will

also be a function of both density and temperature (see, for example, Sect. 8.2).

If we now consider a cloud without a background source ðIn0 ¼ 0Þ, the observed

specific intensity is the Planck intensity for an optically thick source (Eq. 6.20) and the

observed specific intensity is the Planck intensity diminished by the optical depth for an

optically thin source (Eq. 6.21). Therefore, regardless of the value of tn, it will always

to be true that,

In BnðTÞ ð6:23Þ

In Sect. 4.1.1, we found that the specific intensity of any signal, In, can be represented

by a brightness temperature (Eq. 4.4) and similarly the Planck curve can be represented

by the kinetic temperature. Therefore, Eq. (6.19) could be written with In and In0

expressed as functions of TBn and TBn0, respectively, and BnðTÞ could be written as a

6.4 IMPLICATIONS OF THE LTE SOLUTION 195

function of T. As temperature increases, so does the respective specific intensity. Then,

following Eq. (6.23) for the case with no background source, we find,

TB T ð6:24Þ

Eq. (6.24) is a simple, yet important result because it means that, for sources in LTE

without background emission, the observed brightness temperature places a lower

limit on the temperature of the source. The brightness temperature can be determined

for any source by simply inserting the observed value of In into Eq. (4.4). Once TB is

known, the true kinetic temperature must be the same or higher.

6.4.2 Observability of emission and absorption lines

Whether a spectral line is observed in absorption or emission in the presence of a

background source is an important clue as to whether the region in which the line

formed is hotter or colder than the background. This can be determined from

Eq. (6.19), as Example 6.1 outlines.

Example 6.1

Under what conditions will a spectral line be seen in absorption or emission?

We will take a simple case in which a foreground cloud is capable of interacting with the

background signal and therefore forming a spectral line at only one specific frequency

(with a small line width) as Figure 6.3 illustrates. The cloud is perfectly transparent to the

background signal at all frequencies off the line. If the specific intensity at the line

frequency is less than the value of the background, then there is an absorption line as in

Figure 6.3.a, and if the line specific intensity is greater than the background, then an

emission line is seen as in Figure 6.3.b. Taking the absorption case and using Eq. (6.19) for

the frequency of the line centre,

In ¼ In0 etn þ BnðTÞ ð1 etnÞ < In0 ð6:25Þ

where In0 is the specific intensity of the background continuum, BnðTÞ is the Planck

function at the temperature of the cloud, and tn is the optical depth of the cloud at the

frequency of the line centre9. Rearranging this equation and repeating the process for an

emission line leads to,

BnðTÞ < In0 ðabsorption lineÞ ð6:26ÞBnðTÞ > In0 ðemission lineÞ ð6:27Þ

9Note that, since a spectral line has some intrinsic width (see Sect. 9.3) the optical depth varies over the line.


From the Planck function and the definition of brightness temperature, this becomes,

T < TBn0 ðabsorption lineÞ ð6:28ÞT > TBn0 ðemission lineÞ ð6:29Þ

Therefore, if the temperature of the foreground cloud is less than the brightness tempera-

ture of the background source, then the line will be seen in absorption (Figure 6.3.a) and if

the temperature of the foreground cloud is greater than the brightness temperature of the

background source, then the line will be seen in emission (Figure 6.3.b). If there is no

background source, then effectively, TBn0 0 and the line is always seen in emission even

if the cloud is cold10. Also, if T ¼ TBn0, then no line will be seen.

The Sun’s lower atmosphere is an environment within which a rich array of absorption

lines are formed, as can be seen in the Solar spectrum of Figure 6.4. These lines are called

Fraunhofer lines and the fact that they are seen in absorption, rather than emission,

indicates that they are formed in a region that is cooler than the background11. The

structure and properties of the Solar atmosphere are shown in Figure 6.5. and details of

the Solar photosphere are given in Table G.11. The photosphere is a region from just below

the surface (the surface is taken to be where the optical depth is unity12) to a height of about

500 km at which the temperature is lowest. Above the photosphere is the chromosphere

(‘sphere of colour’ after its first identification as a thin reddish ring during Solar eclipses,

see Figure 6.6) which is the region in which the temperature rises again, but hydrogen still

remains predominantly neutral. The chromosphere also contains much activity such as

prominences which are denser features, often loop-like, that are seen on the limb of the Sun

and are related to the magnetic field structure. As Figure 6.5.a suggests, it is in the upper

photosphere and chromosphere that absorption lines will form, the hotter background

radiation being supplied by the Sun’s surface whose spectrum is given by the Planck curve.

Iν Iν0

(b) ν

emission line

Iν Iν0

(a) ν

absorption line

Figure 6.3. In this example, a cloud, such as is sketched in Figure 6.2, is capable of forming asingle spectral line at a frequency, nl. If the line is lower than the background level, In0, it is anabsorption line (a) and if it is higher than the background, it is an emission line (b). Eqs. (6.28)and (6.29) give the conditions for the two cases. Note that the x axis could be plotted as a functionof frequency, wavelength, or velocity (for the latter, see Eqs. 7.3 and 7.4).

10Actually the minimum value is TBn0 ¼ 2:7 K which is set by the CMB (Figure 4.3).11A more accurate comparison would take into account departures from LTE that occur in these regions.12More accurately, a stellar ‘surface’ occurs at the depth from which most photons originate, on average. Thisactually corresponds to an optical depth of t ¼ 2=3 (see Ref. [33] for details). For the Sun, however, the differenceis only 12 km and will be neglected. Sometimes the word ‘photosphere’ is used to denote just this surface, ratherthan the region shown in Figure 6.5.


Figure 6.4. This spectrogram reveals the Sun’s Fraunhofer lines, shown as dark vertical bandssuperimposed on background colour. The wavelength increases from left to right along each stripwhich covers 60 A. There are 50 strips with increasing wavelength from bottom to top, for a totalspectral coverage from 4000 A to 7000 A. An intensity trace across these strips would result in aspectrum similar to that shown in Figure 4.5 centre (with proper calibration of the backgroundbrightness). This image has been created from a digital atlas based on observations taken with theMcMath–Pierce Solar Facility on Kitt Peak National Observatory in Arizona. (Reproduced bypermission of NOAO/AURA/NSF) (see colour plate)

Figure 6.5. Plots of physical conditions in the Sun’s atmosphere as a function of height. A heightof 0 corresponds to the Solar surface which we take to be at t ¼ 1. (a) Temperature as a function ofheight, with labelling showing the parts of the Sun from below the photosphere to the lower corona.(Adapted from Ref. [68]). (b) Density and temperature as a function of height over a larger regionfrom the upper photosphere to the corona. Note the change in hydrogen ionisation state at a heightof about 2000 km called the transition region. This marks the boundary between the chromosphere(hydrogen mostly neutral) and the corona (hydrogen mostly ionised). (Reproduced by permission ofM. J. Aschwanden)


The Sun also provides us with a good example of emission lines. As the temperature of the

Solar atmosphere continues to rise with height, hydrogen becomes more fully ionised,

marking the boundary between the chromosphere and Solar corona. The corona has a typical

temperature of 106 K but can be an order of magnitude higher or more during Solar flares,

which are extremely high energy (up to 1032 erg) outbursts lasting minutes to hours. The

corona extends well beyond the photosphere, as Figure 6.6 shows. With temperatures higher

than the Solar surface, any spectral lines in the corona would be seen in emission above the

black body spectrum from the surface. Since the density is low, however, these emission lines

are weak. Optical emission lines, for example, do not stand out sufficiently in comparison to

the background photospheric planck curve to be seen. Instead they are more easily observed

at positions in the corona outside of the Solar disk where there is no background. At such high

temperatures, however, emission lines of highly ionized species are present and these can

occur at UVor X-ray wavelengths. Figure 8.11, for example, shows a UVemission line image

of a large coronal loop. X-ray emission lines are also observed and can be seen against the

Sun’s disk, since the Sun’s black body photospheric emission is negligible, in comparison, at

X-ray wavelengths. Figure 6.7 shows an emission line from a Solar flare, although note that

other continuum-emitting processes are also present in the spectrum.

Figure 6.6. This Solar eclipse image shows us glimpses of the reddish narrow chromosphere aswell as the extended hot corona with its delicate filaments. The discrete features at the Solar limbare prominences and are reddish because of Ha emission. The optical corona is too faint to be seenin normal daylight and is only visible by eye during an eclipse. (Reproduced by permission ofL. Viatour and the GNU Free Documentation License) (see colour plate)


As in the Sun, the spectra of other stars also provide us with important information about

their physical parameters. Absorption lines, in particular, are observed in the spectra of

other stars, as Figure 6.8 shows. The change in absorption line strength with stellar spectral

type can be understood in terms of a temperature sequence, as quantified in Tables G.7, G.8,

and G.9. An analysis of the strengths of various lines and their ratios further leads to tighter

constraints on the temperature and density of stellar atmospheres (see Sect. 9.4.2). Thus,

absorption lines provide a wealth of information about stars, including the elemental

abundances discussed in Sect. 3.3.4.

6.4.3 Determining temperature and optical depth of HI clouds

As indicated in Sect. 3.4.5, any interstellar cloud that is neutral has all hydrogen

particles in the ground state. Therefore, the only detectable spectral line from hydrogen

in a neutral cloud is from the spin-flip transition of the ground state (see Appendix C.4)

resulting in the l21 cm HI line. The lifetime of the upper state of this transition is very

Figure 6.7. Two X-ray emission features at 6.7 and 8:0 keV are seen in this X-ray spectrum of aSolar flare. Each feature consists of a number of spectral lines that are unresolved, mainly fromFeXXIV, FeXXV, FeXXVI, and some ionised Ni. The underlying continuum spectrum is not from Solarblack body radiation which is very weak at X-ray wavelengths, but rather from optically thinBremsstrahlung and free–bound recombination radiation (see Sects. 8.2, 8.3). This spectrum wasobtained by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) telescope,launched in 2002 (NASA). To the left are data from the RESIK spectrometer aboard the RussianCORONAS-F Solar telescope launched in 2001. (Reproduced by permission of K. J. H. Phillips and theRHESSI team)


long, of order 107 years, so the line is considered to be forbidden, as described in

Appendix C.4, and transitions are more likely to result from collisions (Prob. 6.4a). The

fact that collisions are driving the line excitation means that LTE applies for this line,

though not for the atom as a whole (see Sects. 3.4.4, 3.4.5, and 9.2) so the LTE solution

to the Equation of Transfer may be used. Moreover, the l21 cm line is always observed

in the Rayleigh–Jeans limit (hn kT, Prob. 6.4b). Thus the specific intensity is directly

proportional to the brightness temperature (Sect. 4.1.2) and we can rewrite Eq. (6.19) as,

TBn ¼ TBn0 etn þ T ð1 etnÞ ðwith background sourceÞ ð6:30Þ

TBn ¼ T ð1 etnÞ ðno background sourceÞ ð6:31Þ

300 400 500 600 700

0

2

4

6

8

10

12

14

16

18

20

Wavelength (nm)

Nor

mal

ized

Flu

x(F

λ) +

Con

stan

t

Dwarf Stars(Luminosity ClassV)

M5v

M0v K5v

K0v

G4v

G0v

F5v

F0v

A5v

A1v

B5v

B0v

O5v

Figure 6.8. This plot shows a comparison of spectra for various types of stars of luminosity class,V (see Table G.7 for related stellar data). The shape of the background emission in each case roughlyfollows the shape of the Planck curve corresponding to the temperature of the star’s photosphere.The locations and depths of the absorption lines provide a unique ‘fingerprint’ for the types andamounts of elements that are present. The Balmer lines, Ha ðl656:3 nmÞ, H b ðl486:1 nmÞ and H g

ðl434:0 nmÞ can be easily seen and, in this plot, are strongest at spectral type, A1V. (Reproducedby permission of Richard Pogge, with data from Jacoby, G.H., Hunter, D.A., and Christian, C.A.,1984, ApJS, 56, 257)


where TBn is the observed brightness temperature, TBn0 is the brightness temperature of

the background source, and T and tn are the kinetic temperature (taken to be equal to

the spin temperature, TS, of the gas13, see Sect. 3.4.5) and optical depth of the HI cloud,

at a frequency, n, in the line, respectively. For the sake of completeness and for future

reference, note that, in the case of no background source, Eq. (6.31) reduces to,

TBn ¼ T ðtn 1Þ ð6:32ÞTBn ¼ T tn ðtn 1Þ ð6:33Þ

(using Eq. A.3) which shows more explicitly that the brightness temperature will

always be less than the kinetic temperature (see Eq. 6.24) for a gas in LTE.

If we wish to know both the temperature and optical depth of an HI cloud, a single

measurement of TBn is insufficient. However, if an HI cloud covers a background

source and also extends over a larger region where there is no background source, then

both Eqs. (6.30) and (6.31) can be used. We then can solve these two equations for the

two unknowns, T and tn. Fortunately, nature has provided many such examples, since

most HI clouds we wish to understand are nearby, being in the ISM of the Milky Way,

in comparison to the background sources which are typically radio-emitting quasars in

the distant Universe (see Figure 6.10 for an example). Figure 6.9 shows the relevant

geometry (though note that foreground HI clouds are generally irregular in shape).

When the telescope is pointed at the cloud directly in front of a background quasar

which has a brightness temperature higher than the temperature of the cloud, then an

absorption line against the background continuum is seen, as in Figure 6.3.a. The

emergent brightness temperature in this case can be called, TBnðONÞ. When the

telescope is pointing at the cloud but immediately adjacent to the background source,

then an emission line is seen without any continuum and the emergent brightness

Front view Side view

(a) (b)

Foreground

HI Cloud

telescope pointing “off”

telescope pointing “on”

region of background source

TBn0 TBn

Figure 6.9. Geometry showing how a background radio-emitting source can be used to probe aforeground HI cloud. (a) The emission from a background source of brightness temperature, TBn0,impinges on a foreground HI cloud and emerges at a brightness, TBn on the near side of the cloud.(b) The angular size of the background source is shown with a dashed curve. When the telescopepoints at the cloud directly along the line of sight to the background source, it measures abrightness temperature, TBnðONÞ. When it points on the cloud but away from the direction of thebackground source, it measures TBnðOFFÞ:

13The spin temperature, TS, is often retained in these equations instead of T in case the assumption of LTE breaksdown.


temperature is TBnðOFFÞ. Subtracting Eq. (6.31) from Eq. (6.30) for these two cases

and rearranging yields,

tn ¼ lnTBnðONÞ TBnðOFFÞ

TBn0

ð6:34Þ

Since TBn0 can be obtained by measuring the continuum of the background source just

off the line frequency, the optical depth of the HI cloud can be found. We take the

optical depth here to refer to the value at some frequency in the line. (See Chapter 9 for

more information on line shapes and widths.) Once tn is known, either Eq. (6.30) or

(6.31) can then be used to obtain T.

Although nature has provided many examples of background sources behind HI

clouds, one also needs to be aware of some subtleties. For example, this process will

work only if the temperature and optical depth of the cloud are the same in the ON and

OFF positions. Cloud temperatures do not change dramatically from place to place

(consistent with our LTE assumption), but tn depends on both the density and line-of-

sight distance through the cloud (Eq. 5.9). Both of these quantities may vary with

position and therefore a number of OFF measurements should be made as close as

possible to the ON position and the results averaged. This development also assumes

the presence of only a single HI cloud along a line of sight whereas there may be

several. The fact that clouds are moving at different velocities, however, ameliorates

Figure 6.10. Two images of the same region near the plane of the Milky Way taken at l21 cm. Tickmarks separate 0.5 (about the diameter of the full Moon) on the sky, showing what a large regionof the sky these images cover. (a) Radio continuum emission at a wavelength just off the HI line.The two large structures at the top are HII regions in our Milky Way whereas the myriad point orpoint-like sources are quasars in the distant Universe, one of which is labelled and has a brightnesstemperature of TBn0 ¼ 471 K. (b) HI emission (and some absorption) from neutral HI clouds in ourMilky Way. This image, which is at one particular frequency, n, in the line, has already had thecontinuum subtracted (i.e. it represents TBn TBn0

at every position). The value at the location ofthe absorption feature is 126 K and the average value immediately adjacent to it is 55 K.(Reproduced by permission of The Canadian Galactic Plane Survey (CGPS) team, Taylor, A.R. et al.,2003, AJ, 125, 3145).


this problem somewhat. Since the frequency of the HI line will be Doppler-shifted

(Sect. 7.2.1) by different amounts for each cloud, this helps in treating them as discrete

objects along a given line of sight. Another issue is the possibility of variations within

the spatial resolution of the telescope. For example, the telescope’s beam in the ON

position may have a larger angular size than the background quasar, in which case some

fraction of the beam may be ‘filled’ with a signal that has no background. A more

fundamental problem is that the ISM is known to contain not only HI that is cool

ðT 100 KÞ, fairly dense ðnHI 1 cm3Þ and absorbing, called the cold neutral

medium (CNM), but also hot (of order 8000 K), diffuse ð 0:1 cm3Þ HI, called the

warm neutral medium (WNM). Contributions from the hot HI to the beam affect the

results although, because the hot component is less dense, the cold component has

greater effect. For example, if half of the beam is occupied by the CNM at T ¼ 100 K

and half by the WNM at T ¼ 8000 K, this method will result in a temperature of

T ¼ 200 K (Ref. [81]), an error of a factor of 2.

With these caveats in mind, this method is still a very powerful tool for probing the

physical conditions in HI clouds. More sophisticated analyses that take the hot

contribution into account find that the cold component has temperatures that range

from about 20 ! 125 K with a median temperature of 65 K, with warmer clouds more

common than colder ones (Ref. [47]). Optical depths take on a range of values

depending on the cloud size and density. A real example of HI in the Milky Way is

shown in Figure 6.10.

Problems

6.1 (a) Assume that the Earth’s atmosphere can be approximated by a uniform density

plane parallel slab (e.g. one layer of Figure 2.A.1) and modify Eq. (6.9) so that the specific

intensity, In, is a function of the specific intensity above the atmosphere, In0, the vertical

optical depth, tnðz¼0Þ and the zenith angle, z.

(b) Use the result of part (a) to write an expression for the ratio of observed specific

intensities at two different frequencies, In1=In2

, for a source whose intrinsic spectrum is flat

(i.e. the true specific intensity is the same at all frequencies).

(c) For clear sky optical depths of tblue ðz¼0Þ ¼ 0:085 and tred ðz¼0Þ ¼ 0:060, plot a graph

of Ired=Iblue as a function of zenith angle over the range, 0 z 75 and comment on the

result.

6.2 An Ha emission line of specific intensity, In ¼ 2:7 104 erg s1 cm2 Hz1 sr1 at

the line centre is observed from an ionised gas that is in LTE. Find a lower limit to the

temperature of this gas.

6.3 Indicate whether, in the following cases, a spectral line would be seen in emission, in

absorption, or not at all, and briefly explain why. In each case, the vantage point is the

surface of the Earth.


(a) A region of the Sun’s photosphere of temperature, T ¼ 5000 K, measured in a

direction towards the centre of the Sun’s disk.

(b) A Solar prominence of temperature, T ¼ 5000 K, seen in the chromosphere during a

Solar eclipse.

(c) The Earth’s atmosphere at night.

(d) A gas cloud at a temperature of 2.7 K with no discrete background source.

(e) An HII region in front of a background quasar. The quasar has a brightness

temperature of TB ¼ 100 K.

(f) As in (e) but the quasar has a brightness temperature of TB ¼ 80000 K.

6.4 (a) Compare the collisional timescale for de-excitation of the l21 cm line in a typical

interstellar HI cloud to the mean time for spontaneous de-excitation of this line (Tables 3.1,

3.2, and C.1 may be of help) and confirm that the l21 cm line emission is more likely to be

collisionally, rather than spontaneously de-excited.

(b) Show that the l21 cm line is always observed in the Rayleigh–Jeans limit.

6.5 (a) From the information given in Figure 6.10 (see also Sect. 6.4.2), find the

temperature, T, and optical depth, tn, of the HI cloud at the position of the labelled quasar.

Is the cloud optically thick or optically thin?

(b) Assuming that the depth of the cloud along the line of sight is 5 pc, find the number

of absorptions per cm, an, in this cloud at the frequency of the line.

(c) Assuming that the cloud remains the same density, how deep (pc) would it have to be

for it to become optically thick?

(d) What cloud temperature would be required for an HI emission line to be seen at the

position of the quasar?


7The Interaction of Light with Space

[The] cause of gravity is what I do not pretend to know . . . That gravity should be innate inherent

& essential to matter so [that] one body may act upon another at a distance through a vacuum

[with]out the mediation of any thing else . . . is to me so great an absurdity . . . Gravity must be

caused by an agent acting constantly . . . but whether this agent be material or immaterial is a

question I have left to [the] consideration of my readers.

–Isaac Newton, in letters to Richard Bentley (Ref. [179]).

We now wish to consider how light interacts with space. Matter is not ignored in this

process since it is matter that emits light and also affects the light’s path. However, the

interaction is not one of direct contact, such as we saw for scattering and absorption

previously. If the signal did not intercept a single intervening particle or another

photon, and even if we had perfect measuring instruments that did not alter it, the

signal could still be perturbed from the state it was in when emitted. How can a signal

be altered as it travels through a vacuum? In the next sections, we consider

some possibilities. But first, it is helpful to consider what is meant by space, or rather

space–time.

7.1 Space and time

Einstein’s Special Theory of Relativity is based on the postulate that the laws of physics

are the same and the speed of light, c, is the same, regardless of the speed of the

observer. Indeed, the concept of space–time is a result of the intimate link between

space and time via the constancy of c. His General Theory of Relativity later expanded

these concepts to provide a geometrical description of space–time, its relation to

gravitating masses, and its dynamic character, as we shall soon describe.


A geometrical description of space–time requires the use of some coordinates.

Two points can have the same spatial coordinates, x, y, z, and therefore a distance,

0, between them. However, if two objects are at these coordinates at different times,

then they are separated by a time, t, and will have no interaction with each other. This

seems quite obvious, but the issue is much more acute in cosmology. Because of the

finite speed of light and large distances in the Universe, we see the galaxies where they

were in the past, not where they are now. The more distant the galaxy, the farther in the

past we see it because it has taken longer for the light to reach us. Thus we are separated

from these galaxies by both space and time. It is more useful, therefore, to describe a

separation between events in space–time, rather than a separation between points in

space or between instants in time. A mathematical measure of a distance between

events in some ‘space’ is called a metric. By incorporating the appropriate physics, it is

possible to write metrics for specific circumstances. For example, a metric that

describes a homogeneous (meaning the same at every location), isotropic (the same

as viewed in any direction) and expanding universe is called the Robertson–Walker

metric. This metric is important in the derivation of the Hubble Relation described in

Appendix F. A metric that describes the curvature of space about a spherically

symmetric stationary mass (see below) is called the Schwarzschild metric. The

Schwarzschild metric is used in the development of the relevant equations for the

gravitational redshift (Sect. 7.2.3) and gravitational lenses (Sect. 7.3.1). We do not

present the various metrics, mathematically, in this text (see Ref. [33] for some

examples). However, even without a mathematical description, it is possible to

describe and visualize some of the concepts of relativity. Central to this is the concept

of space–time as an ‘entity’. For the moment, let us consider only space.

The fact that matter moves through space is well known since this is part of our day-

to-day experience. Not obvious, however, is the fact that space can ‘move matter’, that

is, the trajectory of a particle will depend on the specific geometry of the space through

which it travels. Such effects are too small to be measurable on day-to-day human

scales, but are very important cosmologically. The ‘vacuum of space’, even when

devoid of all normal matter, radiation, magnetic or electric fields, still has properties as

well as a residual vacuum energy. Indeed, particles can appear out of this vacuum for

very brief periods of time. These are called virtual particles and cannot be measured

directly. Thus space is not empty, but is an entity that can expand, curve, or distort.

A helpful visualization is to represent three-dimensional space as a two-dimensional

surface that can stretch and deform (Figure 7.1). In this picture, a photon or particle that

moves through three-dimensional space is represented as travelling along the surface of

the sheet. If the sheet is perfectly flat, then two parallel light beams sent out over its

surface will remain parallel indefinitely. This is called flat, or Euclidean space. If the

sheet had some global curvature, then initially parallel beams would either converge

(positive curvature like a sphere) or diverge (negative curvature like a saddle).

On the largest cosmological scales, the best available evidence suggests (Ref. [142],

Ref. [159]) that space is flat, like the regions far from the central depression in

Figure 7.1. This does not mean, however, that space is static. A flat sheet can stretch

outwards and still remain flat as it carries any embedded particles along with it.

208 CH7 THE INTERACTION OF LIGHT WITH SPACE

Similarly, Euclidean space can expand and carry the galaxies with it, a property that is

further described in Appendix F.

On small local scales near any mass, space is not flat but curved, and the curvature is

greatest for the most strongly gravitating masses. This is analogous to a sheet with a

heavy ball at the centre (Figure 7.1). Any photon travelling near the mass will

experience a change in direction as it moves through this curvature. This is called

gravitational bending and will be discussed in Sect. 7.3.1.

Figure 7.1. Conceptualization of three-dimensional space as a two-dimensional surface that canbe curved by a gravitating mass. The two parallel lines at left represent parallel light beams thatremain parallel in flat space. This is what our Universe is believed to be like on large scales. Thecurve represents light that travels near the mass and is deflected. This is what producesgravitational lensing (Sect. 7.3). The two short arrows represent a light ray that leaves the surfaceof the massive object in a radial direction, helping to illustrate the gravitational redshift(Sect. 7.2.3).

7.1 SPACE AND TIME 209

Including the element of time again, a profound consequence of relativity, now

experimentally verified, is time dilation. A photon travelling near a mass is travelling a

greater distance through curved space in comparison to a photon that is farther away

(Figure 7.1). Since a local freely-falling observer near the mass must measure the same

speed of light, c, in comparison to an observer farther away, it follows that a unit of time

near the mass is greater than a unit of time farther away. In a sense, time and space

‘stretch’ together. If the observer near the mass sends out a signal at regular one second

intervals according to his clock, a distant observer will measure the arriving signals as

being greater than one second apart. Each observer’s clocks are advancing ‘normally’

in their own frame of reference, but not when compared to each other. Similarly, the

observer near the mass will age more slowly in comparison to the observer farther

away. A consequence of this fact is that a photon that leaves a mass will experience a

gravitational redshift. This will be discussed in Sect. 7.2.3.

7.2 Redshifts and blueshifts

If a signal is emitted at a frequency, 0, it may be observed at a frequency, , that is

shifted from the emitted frequency. Since ¼ c=, the observed wavelength will also

be altered from the emitted value, the latter called the rest wavelength, 0. If has

increased in comparison to 0, the light is said to be redshifted and if it has decreased,

the light is blueshifted. For a photon, E ¼ h c=, so redshifted and blueshifted photons

have lower and higher energies, respectively.

Since, on cosmological scales, observed wavelengths are redshifted (Sect. 7.2.2),

changes of wavelength are often described by a parameter called the redshift parameter

or just redshift, z, defined by,

z

0

¼ 0

0

¼> z þ 1 ¼

0

ð7:1Þ

Thus, z describes the fractional change in the wavelength. An observed increase in

wavelength corresponds to positive z and a decrease in wavelength to negative z. There

are three possible origins for such a shift, as will now be described.

7.2.1 The Doppler shift – deciphering dynamics

7.2.1.1 Kinematics – the source velocity

As a source moves through space, its velocity, v (the space velocity), can be expressedin terms of a component that is directed towards or away from the observer, calledthe radial velocity, vr, and a component that is in the plane of the sky, called thetransverse velocity, vt (see Figure 7.2a). Then v ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2r þ v2

t

p.


It is the radial velocity that produces a Doppler shift1 as the source moves through

space. A motion away from the observer results in a redshift and a motion towards the

observer gives a blueshift. Note that r is the radial velocity of the source with respect to

the observer. If both the source and observer are deemed to have motions with respect

to some other reference centre or rest frame, then the radial velocity producing the

Doppler shift is the difference in radial velocity components between the source and

observer, each measured with respect to the rest frame2. For example, motions can be

measured with respect to the Sun (as if the Sun were at rest), with respect to an

imaginary point moving in a perfect circle about the centre of the Galaxy at the location

of the Sun (called the Local Standard of Rest, LSR), or with respect to the centre of the

Milky Way.

From the Doppler shift formulae of Table I.1 and Eq. (7.1),

z þ 1 ¼1 þ vr

c

1 vr

c

1=2

z ¼ vr

cðvr cÞ ð7:2Þ

The second expression, which is the approximation for radial velocities much less than

the speed of light, is all that is needed for most cases (Prob. 7.1). This simple relation is

1There is also a Doppler shift for transverse motion when speeds are relativistic. This is due to time dilation andwill not be considered further here.2This is different from the Doppler shift for sound which requires a medium within which to propagate andtherefore has an absolute rest frame. Thus, the source velocity and the observer velocity must be treated separatelywhen dealing with the Doppler shift for sound.

d2

(a)

(b) d1

µ2 µ1 u t

ur

u t

u t

uline of sight

Figure 7.2. (a) An object’s space velocity, v, can be represented as the vector sum of thetransverse velocity in the plane of the sky, vt, and the radial velocity in the line of sight, vr. (b)The transverse velocity can be determined from the proper motion if the distance is known,i.e. vt ¼ m d. A nearby object (at d1 in the figure) will have a higher proper motion than a distantobject (at d2) for the same vt.

7.2 REDSHIFTS AND BLUESHIFTS 211

very powerful because it directly relates an observable quantity, z, to a physical

parameter of the source, r, independent of any other knowledge of the source,

including its distance. All that is necessary is to ensure that some spectral feature,

usually a spectral line (Prob. 7.2) can be identified, as Example 7.1 illustrates.

Example 7.1

The H a line, emitted from a location near the centre of the Andromeda galaxy, M 31, isobserved at a wavelength of ðHÞ ¼ 655:624 nm. Write an expression for the radial velocityof a galaxy, assuming vr c and determine the radial velocity of the Andromedagalaxy.

From Eqs. (7.1) and (7.2), the radial velocity is given by,

vr ¼ 0

0

c ¼

0

c ð7:3Þ

Eq. (7.3) can also be written in terms of frequency,

vr ¼c

c0

c0

" #c ¼ 0

h ic ¼

c ð7:4Þ

We need only one of these equations and, given 0ðHÞ ¼ 656:280 nm from Table C.1, as

well as the above value for ðHÞ, Eq. (7.3) leads to vr ¼ 300 km s1. Thus, the

Andromeda galaxy is approaching us.

The problem of obtaining the transverse velocity is less easily solved. If the object is

relatively nearby, then it may be possible to measure its angular motion in the plane of

the sky over time (Figure 7.2b). This is called the proper motion, , and it is typically

quite small, being measured in units of arcseconds per year. If the distance to the object

is known, then Eq. (B.1) can be used to convert the proper motion to a transverse

velocity (km s1) with proper conversion of units (e.g. vt ¼ md will give vt in km s1

for in rad s1 and d in km). This kind of measurement is only possible for the nearby

stars (Prob. 7.3). For more distant objects, statistical arguments can sometimes be used.

For example, stars in a cluster may be moving with random motions in which case we

might expect vr vt, on average, with respect to the cluster centre. In other cases, the

velocity may be inferred based on geometry. In a worst case scenario, vt is simply

accepted to be unknown and vr places a lower limit on the space velocity. However, for

any arbitrary configuration in the sky, the transverse velocity is likely to be of the same

order of magnitude as vr unless the object happens to be in a ‘special’ geometry of

moving very close to the plane of the sky or line of sight.


7.2.1.2 Dynamics – the central mass

As early as 1619, Johannes Kepler realized that the orbital motions of the planets were

governed, not by the mass of the planet itself, but by the mass of the Sun. In modern

form, Kepler’s third law3 states,

T 2 ¼ 42 a3

G ðM þ mÞ ð7:5Þ

where T is the period of the orbit, G is the universal gravitational constant, and a is the

semi-major axis of the elliptical orbit (see Appendix A.5). Eq. (7.5) indicates that, for

m M, the period depends only on the mass of the object being orbited. Since the

period, in turn, depends on the velocity of the orbiting body and its separation from M,

it is sufficient to measure the velocity and separation of the orbiting object to find the

interior mass. This is a very important and powerful concept in astronomy because it is

relatively straightforward to measure velocities via Doppler shifts (Example 7.1).

Thus, the orbiting body or bodies act like test particles that probe the central mass

about which they move. Moreover, this method can be generalized to include extended

mass distributions, rather than the simple point-like mass of the Sun at the centre of the

Solar System, as we show below.

It is frequently of interest, then, to determine a velocity, not with respect to us, but

with respect to another centre of rest. Restricting ourselves to the simpler case of

circular orbits4 the important quantity is the circular velocity, c, about the centre. For

example, the Earth’s orbital motion is about the Sun and the Sun’s orbital motion is

about the centre of the Milky Way galaxy. Similarly, we might want to consider the

orbital motion of extra-solar planets about their parent star or the motions of stars about

the centre of another galaxy. If the entire external system (central mass plus orbiting

bodies) has some radial velocity with respect to the observer (us), this motion is called

the systemic velocity, vsys.

Rotating thin disks are examples of extended mass distributions for which the orbits

of individual particles can be closely approximated by circles. They are frequently

observed in many contexts in astronomy and have a common geometry. By thin, we

mean that the disk thickness is much less than the disk diameter. A thin disk would

consist of individual particles, each with its own rotational speed, vcðRÞ, at different

radii, R, from the centre. Examples are the Solar System-sized dusty disks around stars

(Figure 4.13), the galaxy-sized dusty disks in elliptical galaxies (Figure 3.22), and hot,

gaseous disks (called accretion disks since their material accretes onto the central

3Kepler’s first law states that the orbits of the planets are ellipses with the Sun at one focus, and the secondlaw indicates that the planets ‘sweep out’ equal areas of the ellipse in equal units of time. The latter pointrefers to the fact that the planets move fastest when they are near perihelion and slowest near aphelion (seeTable A.1).4Over a period of time, elliptical orbits tend to evolve towards circularity, which corresponds to the lowest energyorbit. There are many examples of circular or nearly-circular motions in astronomy.


object) around compact objects like neutron stars and black holes. Spiral galaxies can

also be modelled as thin disks, the individual particles being all those objects making

up the disk such as stars, gas, dust, and nebulae (see Figure 3.3). These include the

nearly face-on galaxies like M 51 (Figure 3.8) and NGC 1068 (Figure 5.5), galaxies of

intermediate inclination like NGC 2903 (Figures 2.11, 2.18, and 2.19) and galaxies that

are edge-on to the line of sight like NGC 3079 (Figure 3.16). The inclination of a

galaxy is measured with respect to the plane of the sky, so that a face-on galaxy has

i ¼ 0 and a galaxy that is edge-on to the line of sight has i ¼ 90. A thin circular disk

looks elliptical in projection and the inclination can therefore be determined by the

geometry of the ellipse, as shown in Figure 7.3,

cosðiÞ ¼ b

að7:6Þ

where b is the semi-minor axis as projected in the plane of the sky and a is the galaxy’s

semi-major axis. The quantities, a and b, can be expressed in linear units (e.g. cm, kpc)

or angular units (e.g. radians, arcminutes), see Eq. (8.1), as long as the same units are

used for a and b.

The geometry of a rotating disk and how it relates to the observed Doppler shifts of

spectral lines is shown for a galaxy disk in Figure 7.4. For any radius, R, along the

major axis5

vr ¼ vsys þ vcðRÞ sinðiÞ ð7:7Þ

At the galaxy’s centre, the circular velocity is zero, so the radial velocity measured at

the galaxy’s centre is just vsys, as Example 7.1 implied. Since i is known from Eq. 7.6,

vcðRÞ can be found for any position, R, along the major axis, provided a spectral line

can be identified and vr determined for that position.

Face-on view ofinclined circular disk

(a) (b)

side view ofinclined circular disk

o oi

b

a

ab

Figure 7.3 (a) A circular galaxy that is inclined to the line of sight will look elliptical inprojection. The galaxy’s true radius is a. When inclined, its apparent semi-major axis is a but itssemi-minor axis is b, as shown. (b) This side view shows the minor axis of the inclined galaxy of (a)and its inclination.

5Off the major axis, the equation is vr ¼ vsys þ vcðRÞ cosðÞ sinðiÞ. The angle, , is measured in the plane of thegalaxy with a vertex at the galaxy’s centre. It is the angle between the major axis and the line between the galaxycentre and the point off the major axis that is being considered.


It is even possible to determine vcðRÞ for our own Galaxy, although the geometry is

somewhat more complicated (and will not be described in detail here) because of our

unique position within the Milky Way (see Figure 3.3). With a rotating disk model, it is

possible to use the observed radial velocity of an object together with a knowledge of

vcðRÞ to obtain the distance to the object. This is an important way of obtaining

distances, called kinematic distances, within the Milky Way6

The development described above for obtaining vcðRÞ for a rotating disk is very

powerful because it allows us to determine the dynamics of galaxies or any other object

for which circular velocities can be derived. If a stable rotational configuration has been

1 1 1

1 2 2 2 2 0 0 0o

At 2: At 2: At 2: i i

ucsin i = o ucsin i ucsin i = uc

uc

ucuc

I I I I1 2 11

22

o o o

Vsys Vsys Vsys Vsys u u u u

(a) Face-on (b) Intermediate (c) Edge-on (d) Unresolved

Figure 7.4 (a) The top image shows a face-on galaxy (i ¼ 0) that is rotating in the plane ofthe sky. Points 1 and 2 at the ends of the disk are marked, as is point o at the centre. Below this,the middle diagram shows a side view, indicating that there is no component of vc projecting ontothe line of sight. The bottom view indicates the spectrum from such a galaxy. Since there is noradial motion with respect to the centre, all points lie at the same radial velocity which is just thesystemic velocity, vsys. (b) As in (a) but for a galaxy of intermediate inclination. In this case, thereis a radial velocity component with respect to vsys so points 1 and 2 are separated from point o inthe bottom spectrum. (c) As in (a) but for an edge-on disk (i ¼ 90). Points 1 and 2 are separatedby the maximum amount in the spectrum. (d) Example of a galaxy of intermediate inclination thatis unresolved spatially. The dashed circle in the top image shows the size of the resolution element.In this case, all points in the galaxy are received at once so the spectrum shows a continuousdistribution of velocities centred on vsys. The maximum values of vr on either side correspond topoints 1 and 2

6The geometry is such that two distances are determined, one corresponding to the near side and one to the far sideof the Galaxy. Thus, there is a distance ambiguity that remains in this analysis. If other independent informationabout the object is available, it can often be brought to bear on the problem so that one of the two distances can bediscarded in favour of the other.


established, this means that the gravitational force, FG can be equated to the centripetal

force, Fc (see Table I.1) which, re-arranging and solving for central mass becomes,

MðRÞ ¼ fR ½vcðRÞ2

Gð7:8Þ

Here, M(R) is the mass interior to the point, R from the centre and G is the Universal

gravitational constant. The quantity, f, is a geometrical correction factor to account for

the fact that the distribution of interior mass may be flattened rather than spherical. For

a spherical mass distribution (like a spiral galaxy with a dark matter halo, see below)

and for a situation in which virtually all mass is contained in a central point (like the

Solar System), f ¼ 1. If we want to obtain the total mass of an object, such a calculation

should be done at the outermost measurable point so as to include as much interior

mass as possible. For values expressed in common astronomical units, the above

equation becomes,

M

M¼ 2:32 105 f

R

kpc

vc

km s1

h i2

ð7:9Þ

where the dependence of mass and velocity on R is taken to be understood.

Eq. 7.9 can be used for any circularly rotating disk consisting of individual freely

orbiting particles, or any single circularly orbiting body whether the interior mass is

observed or not. All that is required is that a spectral line be identified and measured to

obtain v and the distance to the object be known in order to convert an angular distance

along the major axis to R. Note that the spectral line is being emitted by an object

whose own mass does not enter into Eq. (7.9). Thus, as indicated earlier, the moving

object is a tracer of the mass that it orbits. For example, the mass of a black hole can be

found from the accretion disk orbiting it, the mass of an unseen massive planet can be

found from the orbit of the central star about the common centre of mass (a small but

measurable perturbation, Sect. 4.2.2), the mass of a galaxy, including the parts that we

can see as well as any dark matter halo interior to the measured point can be found by

measuring the Doppler shifts of the outermost points on the major axis, as Example 7.2

describes.

Determining the mass distribution of a galaxy can be achieved by measuring the

circular velocity, vc, as a function of R. A plot of vcðRÞ is called a rotation curve and it is

observationally found that vc is approximately constant with R in the outermost parts of

galaxies. This means that the mass continues to rise with R in these regions (M / R,

from Eq. 7.9, when vc constant) even though the light is seen to fall off exponen-

tially. This is one of the strongest arguments for dark matter in and around galaxies. In

the other extreme, in which virtually all mass is contained in the central point, all test

particles ‘feel’ the central mass regardless of their distance. This implies that

vc / 1=ffiffiffiR

p(since M constant in Eq. 7.9). An example of this case is the Solar

System, for which M ¼ M. Substituting a ¼ R and vc ¼ 2R=T in Eq. (7.5) leads to

the same proportionality for vc.


Example 7.2

A spiral galaxy at a distance of 20 Mpc and vsys ¼ 525 km s1 has semi-major and semi-minoraxes of a ¼ 5:00 and b ¼ 2:30, respectively. The 21 cm line of HI at the farthest measurablepoint along the major axis has an observed wavelength of 21:155190 cm: Find the mass ofthe galaxy interior to the measured point.

From Eq. (7.6), the inclination is i ¼ cos1ðb=aÞ ¼ cos1ð2:3=5:0Þ ¼ 62:6. The outer-

most measurable point along the major axis (50) is at a distance (by Eq. B.1 and converting

to kpc and radians) of R ¼ ð20 103Þ ð1:45 103Þ ¼ 29:1 kpc. The radial velocity of

this point, from Eq. (7.3) and using 0 ¼ 21:106114 (Table C.1), is 697:1 km s1. There-

fore, this side of the galaxy is receding with respect to its centre. From Eq. (7.7) and known

values of inclination and systemic velocity, we find a circular velocity of

vc ¼ 193:8 km s1. Finally, adopting a spherical mass distribution (f ¼ 1), Eq. (7.9) yields

M ¼ 2:5 1011M.

For situations in which the whole rotating disk is spatially unresolved, the emission

from all parts of the disk are received in a single resolution element, as shown in Figure

7.4.d. All radial velocities represented in the galaxy are then blended into a single broad

feature as indicated in the spectrum, although the shape may vary depending on the

distribution of material in the disk and inclination. If it is known from some other

measurement that the object is indeed a rotating disk, then it is still possible to obtain

the central mass by noting that the central velocity of the spectrum should correspond

to vsys and the extreme points of the spectrum should correspond to the radial velocity

at the maximum value of R as the figure indicates (Prob. 7.6).

Considering the geometry of a rotating disk is of great use in obtaining the interior

mass over a large range of size scales, from dusty systems around individual stars to

entire galaxies. In principle, however, this kind of dynamical study can be applied to

any gravitationally bound system that isn’t necessarily in the geometry of a thin disk.

The stars in elliptical galaxies (see Figure 3.22), for example, are in a more spherical

distribution that results from individual stellar orbits over a wide variety of angles.

These stars are moving in individual orbits that are dictated by the gravitational

potential of the elliptical galaxy as a whole. A statistical analysis of such stellar

motions then leads to a determination of the mass – both light and dark mass – of

the galaxy. Going to a significantly larger scale, by studying the motions of many

individual galaxies that are in gravitationally bound galaxy clusters (see e.g. Figure

8.8), the mass of the cluster can also be found. Thus, a tiny Doppler shift in the

wavelength of light can lead to a knowledge of light and dark mass, including the

largest known masses in the Universe.

7.2.2 The expansion redshift

As early as 1914, it was realized through the work of the American astronomer, Slipher,

that galaxies preferentially showed redshifts, rather than blueshifts. As more data were


acquired, it became clear that almost all galaxies showed redshifts. Edwin Hubble later

determined the distances to a number of galaxies and, combining his results with

the measurements of Slipher, found that galaxies at larger distances also showed

larger redshifts. This is known as the Hubble Relation, a modern version of which is

shown in Figure F.2. The slope of the curve closest to the origin is called the Hubble

constant, H0.

The Hubble Relation is a direct consequence of the expansion of the Universe (see

Figure F.1). A photon that leaves a distant galaxy must travel through a universe that is

expanding. As a result, the wavelength of the observed photon will be ‘stretched out’ by

this expansion in comparison to its wavelength in the rest frame in which it is emitted.

Unlike the Doppler shift, this redshift does not result from galaxies moving through

space. Rather, the expansion of space itself carries the galaxies with it, consistent with

the properties of space discussed in Sect. 7.1. More distant objects emit light from

regions in which the expansion is greater with respect to us and therefore they have

higher redshifts. Since redshifts are easily obtained but distances are not, the calibrated

Hubble Relation is used to find the distances to galaxies and quasars, assuming that the

galaxy or quasar is far enough away that its redshift is dominated by the expansion of

the Universe rather than its peculiar velocity through space (Example 7.3). If this is

true, then a redshift, z, can be converted into a distance, d, via Eq. F.1 or, for larger

redshifts (z00.1), via Eq. F.16. At redshifts higher than z ¼ 1, a more complex

expression is required (Appendix F.3).

Example 7.3

Estimate a typical distance beyond which a galaxy’s redshift is dominated by the expansion ofthe Universe rather than its peculiar motion through space.

From Eq. (7.2) and Eq. (F.1), we wish to know the distance, d, at which,

c z ðExpansionÞ ¼ H0 d > vrðDopplerÞ ð7:10Þ

We will take the cluster of galaxies, Abell 2218 (see Figure 7.6), as a typical example of the

largest, most massive structures in the Universe and therefore the ones capable of produ-

cing the highest galaxy velocities through space. A typical velocity of a galaxy in this

cluster is vr ¼ 1370 km s1. Using H0 ¼ 71 km s1 Mpc1 (Table G.3), we find

d >19:3 Mpc. In terms of extragalactic sources (those outside of the Milky Way), this is

not very distant. It is about the same distance as the Virgo Cluster, which is the closest

substantial cluster of galaxies to us. Therefore, beyond approximately 20 Mpc, the redshift

is dominated by the expansion of the Universe and peculiar motions through space can be

neglected unless they are unusually high.

The expansion redshift is defined by considering how much the Universe has

expanded between the time the light has been emitted, tem, and the time that it is


observed, tobs. The fractional increase in the wavelength of light due to expansion, z, is

then equal to the fractional increase in the size of the Universe between these two times,

z ¼ aðtobsÞ aðtemÞaðtemÞ

¼> z þ 1 ¼ aðtobsÞaðtemÞ

ð7:11Þ

where aðtÞ is a dimensionless scale factor that describes how the scale (‘size’) of the

universe changes with time. A galaxy with a measured redshift of z ¼ 6, for example,

emitted its light when the Universe was only 1/7th of its current size. In Appendix F we

use this information to derive the Hubble Relation (see Figure F.2) and also to consider

how the observable properties of this relation relate to the energy and mass density of

the Universe that were presented in Sect. 3.2.

7.2.3 The gravitational redshift

Since any mass will distort space–time around it, a photon that leaves the surface of a

mass must emerge from a region of curved space–time. Such a photon is redshifted as a

result. This gravitational redshift is a result of time dilation near the gravitating mass.

Imagine a signal, such as repeated short pulses of light, emitted at regular time intervals

from a region near the mass. A distant observer will measure these time intervals to be

longer than the intervals he would measure in his own rest frame for the same process.

Thus the observed frequency of these pulses ( / 1=t), as measured by the distant

observer, will be lower. If the signal is now a wave of some frequency, rather than

individual pulses, then the observed wavelength will be longer, or redshifted, in

comparison to the rest wavelength.

It is sometimes useful to look at this in a more classical fashion such as would be

done for a particle with a mass, rather than a photon. That is, it takes energy for a

photon to climb out of a gravitational potential well. Since the emergent photon’s

energy decreases during this process, its wavelength must increase. A gravitational

blueshift is also possible for an incoming photon as it gains energy approaching a mass.

An expression for the gravitational redshift is derivable from the Schwarzschild

metric, the result being,

z þ 1 ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1 rS

rÞ

p rS < r ð7:12Þ

where z, measured by a distant observer, is the redshift of a photon that originates at a

distance, r, from the centre of the mass7 and rS is the Schwarzschild Radius, defined by,

rS 2 G M

c2ð7:13Þ

7At r rS, the coordinate system of an external observer breaks down and we do not consider this case further.


The Schwarzschild Radius is the radius of a black hole, i.e. the radius of a region of

space within which the gravity is so strong that not even light can escape8. Any mass, M

that is completely contained within a radius, rS, will be a black hole. (See Sect. 3.3.3 for

information on the formation of stellar mass black holes.)

From Eq. (7.12), it can be seen that the gravitational redshift will not be a strong

effect unless the photon leaves an object from a position, r, that is close to its

Schwarzschild Radius. Any object that is not a black hole has a radius that is larger

than its Schwarzschild Radius (usually much larger), so the gravitational redshift is

negligible for most objects (Example 7.4). However, it can become important for very

compact or collapsed objects like neutron stars and the light-emitting regions surround-

ing black holes (Prob. 7.12, see also Example 8.3).

Example 7.4

Compare the magnitude of the gravitational redshift to the Doppler shift from a Solar-type starthat is receding from us at a velocity of vr ¼ 20 km s1.

From Eq. (7.2), z ðDopplerÞ ¼ vr=c ¼ ð20=3:00 105Þ ¼ 6:7 105. From Eqn. (7.13) for

a 1 M star, rS ¼ ð2 GÞ ð1:99 1033Þ=c2 ¼ 2:95 105 cm. A photon leaves such a star at a

radius, r ¼ R ¼ 6:96 1010 cm, so using Eq. (7.12), z ðGravitationalÞ ¼ 2:1 106,

which is more than an order of magnitude smaller than the Doppler shift.

7.3 Gravitational refraction

Just as a photon which leaves a mass must travel through curved space – time to escape,

a photon from a background source that passes near a mass en route to us must also pass

through curved space – time. If the background light source, the mass and the observer

are sufficiently aligned, then the observer may see an image or images of the source

displaced from its true position in the sky. The mass is called a gravitational lens since

it acts like a convex lens, refracting the light rays towards it.

A variety of image geometries may result, depending on the alignment and distances

of the source and lens as well as the source light distribution and lens mass distribution.

Several examples are shown in Figure 7.5 and Figure 7.6.

7.3.1 Geometry and mass of a gravitational lens

The geometry of a gravitational lens is shown in Figure 7.7a. The light path is actually

curved, but it can be represented by a single bending angle, , defined as the difference

8The Schwarzschild Radius is accurately derived by considering General Relativity. However, a simplified way inwhich to understand the equation is to set the escape velocity from the object equal to the velocity of light, i.e.vesc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2GM=r

p¼ c, from which Eq. (7.13) follows.


between the initial and final directions. It has therefore been drawn as two straight ray

paths in Figure 7.7a with a single bend. The bending angle, , was determined by

Einstein for an isolated lens and can be derived from the Schwarzschild metric.

Assuming a small bending angle, the result is,

¼ 2 rS

b¼ 4 G ML

b c2 ¼ 1:7500

ML

M

h ib

R

h i ð7:14Þ

where b, the impact parameter, is the distance of closest approach to the mass and rS is

the Schwarzschild Radius (Eq. 7.13) of a lens of mass, ML. The left equation provides

the result in radians (all cgs units) and the right in arcseconds for inputs in Solar units.

This result is exactly twice that expected from Newtonian physics9 and catapulted

Einstein to fame when the larger value was confirmed by observations of the displace-

ments of background stars during the solar eclipse of 1919. The potential power of

gravitational lensing as an astronomical tool, however, was realized by S. Refsdal in the

1960s (Ref. [149]).

There are two important differences between a gravitational lens as implied by

Eq. (7.14) and one made of glass (or other refracting material):

No focal length: For a glass lens, incoming rays are bent the least near the lens centre

(zero at the centre) and greatest near the lens edge, allowing light to intersect at a

Figure 7.5 The Einstein Ring, B1938+666. The lensing mass (a galaxy) is at the centre. Contoursshow the 5 GHz radio emission obtained using the Multi-Element Radio-Linked Interferometer(MERLIN) in Great Britain and the grey scale image shows near-IR emission obtained using theHubble Space Telescope (adapted from Ref. [87]). (Reproduced by permission of L. J. King,University of Manchester)

9Space curvature is not expected in Newtonian physics, but the photon has an ‘equivalent mass’ viaE ¼ h ¼ m c2 and could be treated as a small particle to find a deflection angle in a Newtonian model.

7.3 GRAVITATIONAL REFRACTION 221

single point in the focal plane. This is accomplished by machining the surfaces to the

desired curvature. For a gravitational lens, on the other hand, the bending angle is

inversely proportional to the impact parameter, so light is bent less at larger distances

from the lens centre. Thus a gravitational lens has no focal length.

Wavelength independence: The speed of light in a medium depends on interactions

between light and the particles of the medium and such interactions depend on

the wavelength of the light. Thus, for a glass lens, the index of refraction (n ¼ v=c,

Table I.1) is wavelength dependent, blue light being bent more than red light.

However, the speed of light in a vacuum is constant. Therefore Eq. (7.14) has no

wavelength dependence and a gravitational lens is always achromatic. X-rays, IR or

radio waves are all bent through the same angle, provided the background source

emits in these bands and the emission is coming from the same location. Note how

the radio and near-IR emission occurs at the same angular distance from the lensing

galaxy in Figure 7.5, for example.

Figure 7.6 (a) An optical view of the galaxy cluster, Abell 2218, showing the astonishing warpingof space–time via thin arcs that are images of gravitationally lensed background sources. There areover 100 arcs in this cluster which is at a redshift of z ¼ 0:175. The velocity dispersion of galaxiesin the cluster is 1370 km s1 (Ref. [188]). The x axis spans 12700 and the y axis spans 6400.(Reproduced by permission of W.Couch (University of New South Wales), R. Ellis (CambridgeUniversity), and NASA.) (b) A reconstructed mass model over a larger field of view as in (a) fromRef. [1]. The scale is approximately the same as in (a), the crosses marking the positions of the twobrightest regions in (a). The mass distribution, which includes both light and dark matter, is muchmore spread out than the visible galaxies (Reproduced by permission of P. Saha and the AAS)


From the simplified geometry of Figure 7.7a which assumes that both the source and

lens are points, we can calculate the observed displacement of a background object.

Since the bending angle is actually very small, the impact parameter of the curved path

that the light actually follows is about equal to the impact parameter of a path

represented by the straight lines shown. From this geometry, Eq. (7.14) and the

assumption of small angles, it can be shown (Prob. 7.13) that the observed angle of

the image, , satisfies a quadratic equation, known as the lens equation,

2 2E ¼ 0 ð7:15Þ

where,

2E ¼ 4 G ML

c2

1

dL

1

dS

ð7:16Þ

The latter expression contains the information about the lens mass and the source

and lens distances10. The lens equation has two roots corresponding to two image

Figure 7.7. (a) Geometry of a gravitational lens (with angles exaggerated) when both the sourceand lens are points. A point source at S is a distance, dS, and a true angle, , from the observer at O.A point lens, L, of mass, ML, is a distance, dL, from the observer. The light path, shown here by twostraight lines with arrows, is actually a curved path whose bending can be characterized by thegravitational bending angle, . The impact parameter, b, is the closest distance of the light path tothe lens, L. The observer will see an image at an angle, , from L. (b) Face-on view of agravitational lens in which both the source and lens are no longer points. Here a backgroundelliptical galaxy is lensed by a closer spherically symmetric mass distribution. The observer sees thetwo arcs shown. The location of the background galaxy, the lensing mass and its correspondingEinstein Ring are indicated. The point, X, on the source, at an angle, from the lens, has twoimages: Y, at an angle 1 from the lens, and Y’ at 2 from the lens. Adapted from Ref [111].

10For very large cosmological distances, a distance can be defined in several ways. For gravitational lensing, theangular diameter distance should be used (Ref. [149]) which is the ratio of the object’s physical transverse sizeto its angular size (in radians), i.e. the distance implied by Eq. (B.1).


locations,

¼

2 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 þ 4 2

E

qð7:17Þ

Since the value of the square root will always be greater than , there will be one

positive root (which we call 1) corresponding to the image on the same side of the lens

as the source, and one negative (2) corresponding to an image on the other side of the

lens. Both these angles can be measured, provided the lensing mass location is

observed. Then the true position of the source, which is not observable, can be found

from the sum,

1 þ 2 ¼ ð7:18Þ

and E can be determined from the total angular separation between the two images,

1 2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 þ 4 2

E

qð7:19Þ

A special case occurs if the source and lens are perfectly aligned ( ¼ 0). In this case,

1 ¼ 2 ¼ E. There is no preferred plane, and instead of two individual images, a

ring is formed around the lens called the Einstein Ring of radius, E. An astronomical

example is shown in Figure 7.5.

Determining E is very important since this quantity contains the important astro-

nomical information. The distances to both the source and lens can be found from the

Hubble Relation (Appendix F) if their redshifts are measured, the source redshift being

the same as the redshift of any of its images. Then Eqs. (7.16), (7.18), and (7.19) can be

combined to obtain the lens mass,

ML ¼ c2 1 2

4 G

dL dS

dS dL

ML ¼ ð1:22 108Þ 1j2j

dL dS

dS dL

M ð7:20Þ

The first equation is in cgs units with angles in radians, and the second requires in

arcseconds and distances in Mpc. The significance of ML is that it includes all mass,

both light and dark. Thus, gravitational lenses provide a means for measuring even the

mass that cannot be seen. We now have an independent method of obtaining masses,

other than via the Doppler shifts and Newtonian mechanics described in Sect. 7.2.1.2.

In reality, sources and lensing masses will not be point-like. Figure 7.7b shows an

example with an offset background elliptical galaxy of uniform brightness and a

spherically symmetric lens. The background galaxy can be thought of as consisting

of many individual point sources, each of which creates two images as before (see

points X, Y, and Y’ in the diagram). In general, a gravitational lens distorts the image

which, in this case, is clearly no longer elliptical. Because a lens directs light that would

normally not be seen towards the observer (consider the case of the Einstein ring, for

example), the result is an amplification of the signal (see also Sect. 7.3.2). The flux of a


background source will actually be greater as a result of the lensing than it would be in

the absence of the lens.

With a variety of possibilities for background light distributions and lensing mass

distributions, it is not difficult to imagine that rather complex images can result, as

Figure 7.6a illustrates. One must then model the system, putting in various mass

distributions until the observed images are reproduced (see Figure 7.6b). Although

some assumptions may be required, such modelling can place strong constraints, not

only on the total mass of the lens, but on the spatial distribution of its mass as well.

7.3.2 Microlensing – MACHOs and planets

There are situations in which the bending angles are so small that the entire system is

unresolved. The background source images and lens are collectively seen as a point

source, making it virtually impossible to tell that gravitational lensing is occurring in

any static orientation. However, the lensing mass will have a proper motion (Sect.

7.2.1.1) with respect to the background source and observer. As the lens passes

between the source and observer, there will be an increase and then decrease in the

observed flux of the background source due to the amplification of the signal during the

lensing phase. This is called microlensing. The duration of the microlensing event is

approximately the time over which the background source is within the Einstein radius

of the foreground lens (Prob. 7.16). A characteristic light curve results, such as that

shown for one star lensing another star in Figure 7.8.

It is important to be able to identify such a light curve as being due to a microlensing

event and not simply to a variable star or a supernova, both of which show light curves

that increase, then decrease in brightness. Aside from the shape of the curve, it is

possible to separate true microlensing events from other variable phenomena by the

frequency dependence of the curve. The light curves of variable stars and supernovae

result from processes that involve the transfer of photons through matter which has

some opacity. As we have seen earlier (e.g. Sects. 5.4.2, 6.2) opacity is highly

frequency dependent and, as a result, there tend to be differences, frequency to

frequency, in the light curves of variable stars and supernovae. Since microlensing is

achromatic, however, the light curves should be identical at different frequencies. It is

thus possible to identify true microlensing events this way11.

The detection of microlensing relies on the chance alignment of objects at any time,

an event which is intrinsically unlikely. However, modern telescopes, especially

robotic ones with customized software, have the ability to monitor millions of objects

and routinely extract the lensing candidates. In addition, if fields are chosen for which

there are many possible background objects, then the statistics are much better. These

approaches have been taken in a variety of experiments. For example, microlensing

events have been observed when MACHOs in the halo of the Milky Way (see Sect. 3.2)

11In practise, the process is somewhat more difficult because each pixel receives the light of many stars. Thus, theproblem is approached statistically.


pass in front of the Large Magellanic Cloud which is a nearby galaxy. It is studies like

this that have led to the conclusion that there aren’t enough MACHOs to account for the

dark matter that is expected to be in the Milky Way’s halo (Ref. [4], Ref. [3], Ref. [5]

and Sect. 3.2). Figure 7.8 shows another exciting application of microlensing. The

main light curve results from a background star in the bulge of the Milky Way being

lensed by a foreground star. However, a small perturbation on the right of the curve,

blown up in the right inset, shows another peak, this time due to a planet only 5.5 times

the mass of the Earth, that orbits the lensing star. Thus, microlensing is now detecting

planets with masses approaching that of the Earth, extending the range of possible

detections beyond those that can be explored by other methods (see Sect. 4.2.2).

7.3.3 Cosmological distances with gravitational lenses

Gravitational lensing on very large cosmological scales can also provide independent

information on the distance to a source for the special case in which the source brightness

is varying with time. An example is when the background source is a quasar, most of

Figure 7.8 Characteristic light curve of a microlensing event. Both the lens and the backgroundsource are stars in our Milky Way. The background star is a G4 III giant star at a distance of 8.5 kpc.The lensing star is an M dwarf of mass, M ¼ 0:22M, at a distance of 6.6 kpc. The left inset showsthe light curve, as monitored over 4 years time. The small bump on the right, blown up in the rightinset shows the smaller peak due to a planet, called OGLE-2005-BLG-390Lb, orbiting the lensingstar. The planet’s mass is Mpl ¼ 5:5M and is a distance, r ¼ 2:6 AU from its parent star. This is thethird exoplanet discovered using microlensing techniques. The discovery was made by aninternational team (Ref. [13]) involved in three microlensing projects: Probing Lensing AnomaliesNETwork (PLANET) þ RoboNet, Optical Gravitational Lensing Experiment (OGLE), and MicrolensingObservations in Astrophysics (MOA). (Reproduced by permission of ESO/PLANET/RoboNet, OGLE,and MOA)


which show time variability. When the source is offset from the lens centre and

two images at 1 and 2 are seen, one on either side of the lens (see Figure 7.7),

the path length on one side of the lens will be different from the path length on the

other side. A change in brightness will therefore be observed in the image that has the

shorter path length before the image that has the long path length. The time delay

between the brightness change in the two images is related to the path difference. With

measurements of 1 and 2 and a knowledge of the lens distance, the source distance, dS,

can be determined. Quasars are in the distant Universe where distances are normally

determined by using the Hubble Relation (see Figure F.2), so a method like this is

very important in providing another measure of distance. Since the redshift of the quasar

can be measured (it is the value obtained from the image), this technique has been used

to provide an independent determination of the Hubble constant, H0, with reasonable

results. However, uncertainties about the distribution of mass (which must include

light and dark matter) in the lens has meant that the effectiveness of this technique is

currently limited.

7.4 Time variability and source size

A simple, but powerful argument related to fluctuations in the brightness of a source

has proven to be extremely useful in constraining source sizes which otherwise would

be too small to measure (see Figure 7.9). The size of the region that is variable must be,

d c t ð7:21Þ

where t is the timescale of the variability. This condition is a requirement of

causality. If it were not the case, then the light that is emitted from a varying source

at point A (see figure) would not be able to travel across the source to point B in a time,

t. Point B, on the other side of the source, would therefore not ‘know about’ the

variation at A so could not be in synchronicity with it. If points on a source are varying

independently, then no global variation would be detected. This is similar to arguments

presented earlier as to why planets shine and stars ‘twinkle’ (see Sect. 2.3.4). If

variability is observed, then Eq. (7.21) places a limit on the size of the varying source.

A B AB

t t+∆t

Figure 7.9 The size of a variable source must be less than the light travel time across it. At time,t (left), the source is dim and, at a time, t þ t (right), the source has brightened. All parts of thesource can vary together, provided, d c t.

7.4 TIME VARIABILITY AND SOURCE SIZE 227

Some sources are variable over a variety of timescales. It can be seen from Eq. (7.21)

that the shortest measured timescale puts the tightest restrictions on source size. X-ray

variability is particularly important because the fact that X-rays are observed indicates

that the source is associated with high energy phenomena. X-rays tend to be emitted

near very compact objects that have high gravitational fields, such as white dwarfs and

neutron stars, or active galactic nuclei that are thought to be powered by supermassive

black holes (e.g. Sect. 5.4.2). The shortest variability timescale then provides a limit on

the size of the emitting object (Prob. 7.17). If information about the mass of an object is

available from dynamics (Sect. 7.2.1.2) and information about the size of the object is

available from time variability, then strong constraints can be placed on the type of

object that is present, even if it is spatially unresolved.

Problems

7.1 How high would the radial velocity of an object have to be before the low velocity

approximation for the Doppler shift differs from the accurate expression (see Eq. 7.2) by

more than 10 per cent? (The solution may be obtained ‘by hand’ or via computer algebra

software.)

7.2 In principle, any spectral feature for which the rest wavelength, 0, is known could be

used to measure the radial velocity, vr, of an object. Let us take the peak of the Planck curve

of a star as the ‘spectral feature’. The star has a radial velocity of vr ¼ 50 km s1 and we

assume that its temperature, T0 ¼ 5800 K, has been determined by some other method.

(a) Find the rest wavelength, 0, for the peak of the Planck curve of this star and the

observed, Doppler-shifted wavelength, , of this peak.

(b) Find the equivalent Doppler-shifted temperature, T corresponding to the new peak, .

(c) Compute the specific intensity of the star at the Doppler-shifted wavelength, , if

there were no radial velocity (i.e. BðT0Þ). Compute the specific intensity of the moving

star at the same wavelength (i.e. BðTÞ). What is the change (in per cent) in specific

intensity at the wavelength, , as a result of the star’s motion?

(d) Assume that your measuring instruments can detect changes of about 3 per cent or

larger and indicate whether the change indicated in part (c) could be detected.

(e) Compute the Doppler shift, , of the H line for the same star. If the width of the

H line is ¼ 0:4A, can the H Doppler shift be easily measured or not?

(f) Comment on the suitability of the Planck curve peak, in comparison to a spectral

line, for measuring the radial velocities of stars.

7.3 How close does a star, moving with vt ¼ 20 km s1, have to be in order for its proper

motion to be ¼ 100 yr1 (Sect. 7.2.1.1)? Express the result in parsecs and in units of the

distance of the Sun from the centre of our Galaxy (see Figure 3.3). Do you expect proper

motions to be used to obtain vt for a large fraction of the stars in the Galaxy?


7.4 From the information related to the following test particles, find the mass of the

central object (M):

(a) The Earth in its orbit about the Sun.

(b) A roughly spherical cluster of galaxies for which the outermost galaxies are

approximately 1 Mpc from the centre of mass and the maximum measured radial velocities

are 300 km s1 with respect to vsys of the cluster.

(c) A globular cluster of stars for which the average radial velocity with respect to sys is

vr ¼ 2:8 km s1. The radius corresponding to this average radial velocity is 30 pc. Ignore

star-star interactions and assume that the cluster can be represented by a spherical

gravitational potential.

(d) A white dwarf orbiting an unseen object. The radius of its orbit is 10 AU and the

circular velocity is 21 km s1.

7.5 Refer to Example 7.2 and, using the information given in Figures 2.11 and 2.20,

determine the mass of NGC 2903.

7.6 A galaxy’s nucleus harbours a supermassive black hole of mass, M ¼ 108 M,

surrounded by an edge-on accretion disk that is spatially unresolved at X-ray wavelengths.

The full width of a spectral line of iron, whose central energy is Eline ¼ 6:4 keV, is due

entirely to disk rotation and is 2:0 keV.

(a) What is the maximum circular velocity of this disk?

(b) What is the distance of the accretion disk, at its maximum velocity, from the centre of

the black hole (AU)? Should this correspond to the inner edge of the accretion disk or the

outer edge?

(c) Express the radius of part (b) in units of Schwarzschild Radii.

7.7 (a) Write an expression for the relative error in dL that results from using Eq. (F.1) for

the Hubble Relation instead of Eqn (F.16) and evaluate it for the values of q0 used in the two

curves shown in Figure (F.2). The expression will be a function of z.

(b) Determine this relative error for the two values of q0 for (i) a galaxy with recessional

motion of c z ¼ 5000 km s1, (ii) a galaxy at z ¼ 0:1 and (iii) a quasar at z ¼ 0:6.

7.8 For z < 1 and jq0j < 1, verify that the equation for the comoving coordinate, Eq.

(F.10), results from Eq. (F.9). Note that, in this regime, any terms including ðt0 tÞ3or

higher orders can be neglected.

7.9 (a) Starting with the definition of redshift (Eq. 7.1), express the quantity, ðz þ 1Þ, in

terms of the frequency of a wave, rather than its wavelength and, with this result, show that

Eq. (F.14) follows.

(b) Supernova 1995 K, at a redshift of 0.479, was observed to have a light curve that is

time dilated (Ref. [97]). If nearby supernovae of the same type normally show a light curve


width (full width at half maximum) of about 25 days, what was the width of the light curve

of SN1995K?

7.10 (a) Starting with Eq. (F.16), derive an expression for the slope of the curve of the

Hubble Relation shown in Figure F.2 (i.e. dðczÞ=dðdL)) as a function of z. Confirm that the

slope reduces to H0 for z!o.

(b) Measure the slope directly from Figure F.2 (dark solid curve) at two values of z and

estimate H0 and q0 using the expression from part (a).

(c) Discuss the reasons for any differences between the results of part (b) and the values

of q0 and H0 that have actually been plotted (see caption).

7.11 For a flat universe in which there is a cosmological constant, there is an early time at

which the matter term dominates the behaviour of the expansion and a later time at which

the expansion is dominated by . For the epoch at which the energy densities of each

component are equal (M ¼ ), find numerical values for the following (assume

H0 ¼ 71 km s1 Mpc1; M0 ¼ 0:3, 0 ¼ 0:7; and q0 ¼ 0:55):

(a) the density parameters, M and , (b) the deceleration parameter, q, (c) the

cosmological constant, , (d) the Hubble parameter, H (km s1 Mpc1), (e) the density,

, (f) the scale factor in comparison to the current value, a=a0, (g) the redshift, z, and (h) the

lookback time, t0 t (Gyr).

7.12 Repeat Example 7.4 for a neutron star of mass, M ¼ 2 M, and radius, rns ¼ 20 km.

7.13 Derive the lens equation (Eq. 7.15) from the geometry of Figure 7.7 and Eq. 7.14.

(Recall that tan ¼ , for small angle, .)

7.14 (a) Write the Einstein Ring angle equation (Eq. 7.16) for a case in which the

background source is very distant in comparison to the lens. Express the result in units

of arcsec with constants evaluated, for input masses in M and distances in pc.

(b) Using the result of part (a), compute E (arcseconds) for the following cases: (i) the

Sun, (ii) a neutron star of mass, 1:5 M, at a distance of 60 pc, (iii) a galaxy of mass,

1011 M at a distance of 10 Mpc, and (iv) a cluster of galaxies of mass, 1014 M at a

distance of 20 Mpc.

(c) Using the same equation and the information given about Abell 2218 in Figure 7.6a,

make a rough estimate of the mass of the bright subcluster visible on the right hand side of

Figure 7.6a.

7.15 (a) Consider four very distant background quasars that are in the true configuration of

a square on the sky of length 200 on a side. One of these quasars is perfectly aligned with

the centre of a foreground supermassive black hole of mass Mbh ¼ 4 109 M which is at

the core of a galaxy of distance, dL ¼ 1 Mpc. Plot the images of the four stars that result

from being gravitationally lensed by the black hole. The result of Prob. 7.14a may be

useful.


(b) Suppose these four quasars instead delineated the four corners of a uniform bright-

ness background source. Sketch on the plot the resulting images.

7.16 (a) Write an expression for the duration of a microlensing event, tL, in terms of the

transverse velocity of the lens, vt, the distance to the lens, dL, and the Einstein radius, E.

(b) Beginning with the result of (a), re-express tL (in days) in terms of

dL ðkpcÞ; dS ðkpcÞ; ML ðMÞ; and vt ðkm s1Þ.

(c) Evaluate tL (days) for a lensing mass which is a 0:5 M white dwarf in the Milky

Way halo a distance of 5 kpc away with a transverse velocity of 220 km s1. The back-

ground star is in the Large Magellanic Cloud a distance 50 kpc away.

(d) From the information given in the caption of Figure 7.8, evaluate tL (days) for the

lensing star and compare the result to the observed curve duration. Assume that the

transverse velocity (net of background star, lens and observer) is 220 km s1. Comment

on the result.

7.17 An X-ray binary system consists of a normal star and either a white dwarf, neutron

star, or black hole companion, in which material from the star is accreting onto the

collapsed object. In some X-ray binaries, X-ray variability on timescales as short as a

millisecond have been observed. If the timescale is this short, what is the maximum size of

the source (km)? Is it possible to rule out any of the types of companions for such sources?


PART IVThe Signal Emitted

We have seen how a signal can be altered by our instruments, our atmosphere, and the

matter and space that it encounters en route to our detectors. Each of these steps

provided important information about the signal, about the intervening matter along

and near its path, and even about the large-scale structure of our Universe. It is now

time to turn our attention to the source, itself. How is the original signal actually

emitted? This is an important question because, encoded in the signal, is information

about the properties of the source. An understanding of the various processes involved

in the emission of light will provide us with the keys to unlock these secrets.

Fortunately, there are only a limited number of ways in which charged particles

couple to electromagnetic radiation. This means that, although the number of the

objects in the Universe is incomprehensibly large, there are actually only a small

number of light-emitting mechanisms that are responsible for all of the cosmic

radiation that we see. Some of these mechanisms operate only at very high energies

and so are relegated to a subset of exotic objects. Others are more common and

widespread. Generally, emission processes fall into two categories: continuum radia-

tion, emission that occurs over a broad spectral region, and monochromatic or line

radiation, emission that occurs at a discrete wavelength with a narrow width in

frequency. It is these processes that we now wish to consider.

8Continuum Emission

A scientist must also be absolutely like a child. If he sees a thing, he must say that he sees it,

whether it was what he thought he was going to see or not.

– So Long, and Thanks for all the Fish, by Douglas Adams

Continuum radiation is any radiation that forms a continuous spectrum and is not

restricted to a narrow frequency range, the latter applying to quantum transitions in

atoms and molecules. By emission, we mean a process that starts with matter and ends

with the creation of a photon. This could involve an interaction between a charged

particle and another charged particle, a charged particle and an electric or magnetic

field, or a charged particle with another photon, the end result being a photon.

Following the symbolism of Sects. 5.1 and 5.2, emission is represented as: matter !(matter or photon or field) ! photon. Thus, even though scattering can be thought of as

absorption and re-emission (see Sect. 5.1), the starting point is a photon so we do not

consider it in this chapter1.

Some commonly observed astrophysical continuum processes are included in this

chapter, an important exception being black body radiation which was encountered in

Chapter 4. The goal, for each type of continuum, is to obtain the intensity and spectrum

of the signal, a process that involves determining the emission coefficient, jn (intro-

duced in Eq. 6.2), and/or the absorption coefficient, Eq. (5.8), together with the

appropriate solution of the Equation of Radiative Transfer (see Sect. 6.3) to find the

specific intensity, I . Other quantities such as flux density, luminosity, etc. follow from

I , as described in Chapter 1. To have an expression for I is to have an essential

astrophysical tool since it means that properties of the emission, and indeed of the

source itself, can be characterized. To this end, it is useful to note that, for a uniform

source, f and L will have the same spectral shape as I .


1A decision as to ‘starting point’ is really a decision as to choice of rest frame. See additional comments inSect. 8.6.

8.1 Characteristics of continuum emission – thermal andnon-thermal

For continuum emission, it is helpful to remember that an accelerating charged particle

that is not bound to an atom or molecule will radiate. Since we want to know the

spectrum (a function of frequency, ), it is also helpful to recall that frequency is

inversely proportional to time. For example, an oscillating electron (Appendix D.1.1)

will emit radiation at a frequency ¼ 1=, where is the period of oscillation; an

electron that has a brief ‘collision’ with an ion of duration, t, will emit a photon

within some bandwidth B 1=t. Thus, short timescales are related to high frequen-

cies. In any gas in which there are free charged particles, there are many particles, many

accelerations, and many timescales involved. Emission over a range of frequencies – a

continuous spectrum – will result. As we will see, this range can be quite large,

spanning many orders of magnitude in frequency-space. The Planck curve (Sect.

4.1), for example, takes on all frequencies between zero and infinity, although emission

at some frequencies is much more probable than at others. Thus a characteristic of

continuum emission from astronomical sources is that it is broad-band2.

Other characteristics of continuum emission depend more specifically on the type

of radiation being considered. These fall into two categories – thermal emission and

non-thermal emission – and a few comments can be made for each.

Thermal emission is any process for which the observed signal is associated with a

system whose states are populated according to a Maxwell–Boltzmann velocity dis-

tribution Eq. (0.A.2). Since this distribution defines the kinetic temperature T, for all

thermal emission, I may be a function of various parameters but it will certainly be a

function of the temperature, T. The emitted radiation must therefore be related to

random particle motions in the thermal gas. This can be achieved if the particles interact

with each other via collisions. As indicated in Sect. 3.4.1, for Coulomb interactions, a

collision is an encounter that is sufficiently close that it alters the trajectory of a charged

particle. For a collisional process, we can apply the LTE solution to the Equation of

Radiative Transfer (Eq. 6.19, see also Sect. 3.4.4) which, as we found in Sect. 6.4.1,

implies that TB T Eq. (6.24). In astrophysics, we rarely see temperatures above

1078 K, so an observed brightness temperature higher than this range is a clue that

the process is likely non-thermal (see below). Therefore, a characteristic of thermal

emission is low brightness temperatures. Thermal emission is also intrinsically unpo-

larized. Since there is no particular directionality to velocities in a thermal gas, there is

no particular directionality to the resulting radiation. If polarization is observed, then

either some other process is polarizing the signal after it is emitted (e.g. scattering by

electrons or dust, see Appendix D) or else the emission is non-thermal.

Non-thermal emission is everything else. If I is independent of T, then the process is

non-thermal. Examples are when the emission depends on the acceleration of a charged

particle in a magnetic field or an electric field, or when the emission depends on

2See, however, comments in Sect. 8.5.1.

236 CH8 CONTINUUM EMISSION

collisions with other particles whose velocities are not Maxwellian (for example, a

power law). Depending on the details of the specific emission process, the brightness

temperature could be low or high and there may or may not be polarization of the

resulting signal. We now examine these processes in more detail.

8.2 Bremsstrahlung (free–free) emission

Bremsstrahlung, which is derived from the German words for ‘braking’ and ‘radiation’,

occurs inanionizedgaswhenafreeelectrontravels throughtheelectricfieldofapositively

charged nucleus. The electron is ‘braked’ by the electrostatic attraction of the nucleus,

feeling a Lorentz force, ~Fe ¼ e~E (for zero magnetic field, Table I.1) as it passes by

(Figure 8.1). The electron is accelerated (its direction is changed) by this ‘collision’ with

the nucleus, and will therefore radiate. The process can also be thought of as the scattering

of electrons in the electrostatic field of the nucleus. The resulting radiation is equivalently

referred to as free–free emission. It is the inverse of the free–free absorption process

discussed in Sect. 5.2 in which an electron absorbs a photon in the vicinity of a nucleus.

8.2.1 The thermal Bremsstrahlung spectrum

Derivations of thermal Bremsstrahlung emission can be found in Refs. [143], [101] and

[79] and will not be repeated here. However, it is useful to offer a more qualitative

description of this process in order to see how the functional dependences result.

One first considers a single electron in an encounter with a single nucleus. Given the

mass difference between these particles, the nucleus can be considered at rest. Since we

x

Electron photon

Nucleus

y

V

r

b

Figure 8.1. Geometry showing thermal Bremsstrahlung (free–free) emission resulting from anelectron travelling by a nucleus of charge, Ze, where Z is the atomic number. The heavy curve showsthe electron’s path. An instantaneous velocity vector is shown for the electron at a position,~r, aswell as an x; y coordinate system and the impact parameter, b, which is the distance of closestapproach.

8.2 BREMSSTRAHLUNG (FREE–FREE) EMISSION 237

are dealing with a free–free process, the electron must not recombine with the nucleus.

Rather, the kinetic energy of the electron, Ek ¼ ð1=2Þmev2, must be greater than the

potential energy at the distance of closest approach to the nucleus,EP ¼ Z e2=b, where b

(as we saw with gravitational lenses, Sect. 7.3.1) is the impact parameter and Z is the

nuclear charge (atomic number). For a large range of astrophysical conditions, the change

in angle is small and the change in electron speed is negligible. The energy of the emitted

photon is then much less than Ek of the electron (Prob. 8.1) and the interaction can be

treated as an elastic collision (Sect. 3.4.1) of duration,t b=v. The power radiated by

an accelerating charged particle is given by the Larmor formula (see Table I.1) which can

be integrated over the duration of the encounter and converted to a dependence on

frequency, , instead of time3. With the aid of some geometry and a knowledge of

scattering angles for elastic collisions between charged particles, the result is an expres-

sion for the energy radiated by an electron of speed, v, and impact parameter, b.

The next step is to consider the radiation from an ensemble of particles that are all

interacting with nuclei in this fashion. For many electrons at a single fixed velocity, a

variety of impact parameters are possible so an integration over impact parameter is

required. A reasonable minimum value, bmin, is one at which the electron would be

captured, and a maximum, bmax, would be the value at which the perturbation is so small

that virtually no change in trajectory results. In a real plasma, corrections to these limits

are required, but they are helpful in understanding how the integration over impact

parameter might proceed. Electrons that travel closest to the nucleus will experience the

shortest duration interactions. Therefore, bmin determines the upper limit to the emitted

frequency. Since the electron cannot come arbitrarily close to the nucleus, there should

be a fairly sharp emission cutoff at high frequency for electrons of a given velocity.

The final step is to integrate over a distribution of electron velocities. Any velocity

distribution that is thought to be present could be introduced at this point (e.g. a power

law distribution or even relativistic velocities). However, we are here considering the

most commonly considered Bremsstrahlung emission in astrophysics, that of thermal

Bremsstrahlung which requires the Maxwellian velocity distribution Eq. (0.A.2). The

final result is the free–free emission coefficient (cgs units of erg s1 cm3 Hz1 sr1),

j ¼ 8

3

2

3

1=2Z2e6

me2c3

me

kTe

1=2

ni ne gffð;TeÞ eð hkTe

Þ ð8:1Þ

¼ 5:44 1039 Z2

Te1=2

ni ne gffð; TeÞ eð h

kTeÞ ð8:2Þ

where ni, ne are the ion and electron densities, respectively, Te is the electron

temperature, and Z is the atomic number. The quantity, gffð;TeÞ, is a correction factor

(for example, it includes corrections to bmin for quantum mechanical effects) called the

free–free Gaunt factor4 and is a slowly varying function of frequency and temperature.

The Gaunt factor takes on different functional forms depending on temperature and

3A proper treatment requires the application of a mathematical function called a Fourier Transform. A Fourieranalysis allows one to relate processes in the time domain to the observed frequency domain.4This quantity is sometimes designated with a bar, gff , because it has been averaged over a velocity distribution.


frequency regime (e.g. see Ref. [28]) and is plotted for a wide range of parameters in

Figure 8.2. As indicated when Kramers’ Law for absorption was introduced (Eq. 5.17),

the value of the free–free Gaunt factor is close to unity.

Since the shape of the thermal Bremsstrahlung spectrum follows that of the emission

coefficient for optically thin radiation (see Eq. 6.22 and below), it is worthwhile

pausing to consider the frequency dependence of j (Eq. 8.1 or 8.2). At very high

frequencies (specifically, h > k Te), the exponential term ensures that the emission

coefficient cuts off rapidly, that is, electrons of (most probable, Eq. 3.8) energy, kTe,

cannot emit photons of energy h > kTe. If this cutoff frequency can be observed, then

the temperature of the gas can be determined without the requirement of any other

measurement (Prob. 8.2). Aside from this high frequency cutoff, the only frequency

dependence is within the Gaunt factor. Therefore, the spectral shape of the Gaunt factor

gives the spectral shape of the emission coefficient up to the cutoff. The Gaunt factor

shows little variation with frequency for an object of given temperature and, as we shall

see below, the optically thin thermal Bremsstrahlung spectrum is a flat spectrum.

The emission coefficient, j , tells us the emissive behaviour of a pocket of gas

without allowance for its internal absorption. In order to derive the full spectrum, I , we

also need an expression for the absorption coefficient, , as required by the Equation

of Radiative Transfer (Eqs. (6.3) and (6.4)). In the absence of a background source, the

absorption coefficient tells us how much self-absorption is occurring in the gas, that is,

how effectively the gas absorbs its own radiation. Since the thermal Bremsstrahlung

mechanism arises from a collisional process, the LTE equations can be used (see

Sect. 3.4.4). This means that the absorption and emission coefficients are related via the

Frequency (Hz)

g_f

f

T=1.58E9

1.58E81.58E71.58E6

1.58E5

1.58E4

1.58E3

1.58E2

T=1.58E5(g_ff + g_fb)

10

9

8

7

6

5

4

3

2

1

0

1E+08 1E+10 1E+12 1E+14 1E+16 1E+18 1E+20

Figure 8.2. Value of the free–free Gaunt factor, gffð; TeÞ (taking Z ¼ 1) for a wide variety ofconditions. Each curve is labelled with a temperature. Dark solid curves are accurate values takenfrom Ref. [166]. The three shorter curves at the left are the radio wavelength approximations givenby Eq. (8.11). The grey curve is the combined free–free plus free–bound Gaunt factor for thetemperature, 1:58 105 K from Ref. [28]. Therefore, gfbð; TeÞ for this temperature is representedby the excess above the smooth curve of gffð; TeÞ below it.


Planck function, BðTÞ ¼ j= Eq. (6.18), so and therefore the optical depth, ,

(Eqs. 5.8 and 5.9) can be immediately obtained for a line of sight, l. Inserting Eq. (8.1)

for j,

¼Z

dl ¼Z

jBðTÞ

dl

¼ 4 e6

3 me h c

2

3 kme

1=2

Te1=2 Z2 3 ð1 e

hkTeÞ gffð;TeÞ

Zneni dl ð8:3Þ

ð3:7 108ÞTe1=2 Z2 3 ½1 e

hkTe gffð;TeÞ EM ð8:4Þ

Ref. [143], where ne ni is assumed5 and, since the temperature and Gaunt factor

usually do not change by large factors along a line of sight for any given cloud, these

quantities are taken outside of the integral. The quantity, EM, is called the emission

measure, defined by,

EM Z

ne2 dl ne

2 l ð8:5Þ

The emission measure6 contains information about the number density of particles

along the line of sight. The approximation of Eq. (8.5) is applicable if the density is

approximately constant over l. However, even if the density varies, the emission

measure still provides a useful approximation of the density.

The frequency dependence of the expression for in Eq. (8.3) is of interest. Since

there is little emission when h k Te, we restrict our interest to the regime,

h k Te. Here exp½ðh Þ=ðk TeÞ 1 ðh Þ=ðk TeÞ Eq. (A.3) so the term in

parentheses / . Since the Gaunt factor has only a weak dependence on frequency, it

can be considered roughly constant, so / 2. This means that the opacity of the

cloud increases at low frequencies.

The final spectrum, in the absence of a background source, is obtained by inserting

the Planck Function (Eq. (4.1)) and the above expression for optical depth (Eq. (8.3))

into the LTE solution of the Equation of Transfer Eq. (6.19) which, due to the

complexity of the expression, we leave in its original form here,

I ¼ BðTeÞ ð1 eÞ ð8:6Þ

Results for two different plasmas, one applicable to 104 K gas from an HII region

and one applicable to 106 K gas in a cluster of galaxies, are shown in Figure 8.3.

5He has an ionization energy of Ei ¼ 24:6 eVand so is ionized only in a ‘harder’ radiation field than hydrogen forwhich Ei ¼ 13:6 eV. If HeII is present, it contributes only one electron per ion, consistent with our assumption.HeIII, which contributes two electrons per ion, is negligible in HII regions (Ref. [145]). However, if the situationarises that He is fully ionized, then ne=ni ¼ 1 þ 2NHe, where NHe is the fractional abundance, by number, of He.For Solar abundance, NHe ¼ 0:085 (Sect. 3.3.4), so a ‘worst case’ error made by this assumption is 17 per cent.6Sometimes the Z2 of Eq. (8.4) is included in the definition of emission measure.


The flat nature of the thermal Bremsstrahlung spectrum is quite evident in this

figure.

As we did in Sect. 6.4.1, we can look at the optically thick and optically thin part of

the spectrum. In the absence of a background source see Eqs. (6.20) and (6.22)

I ¼ BðTÞ ð 1Þ ð8:7ÞI ¼ j l ð 1Þ ð8:8Þ

In Figure 8.3, Eq. (8.7) applies to the region where the spectrum turns down at low

frequency, becoming that of a black body. In this limit the measured specific intensity is

a function only of temperature. As the frequency increases, the optical depth decreases,

and the spectrum then departs from the Planck curve. Eq. (8.8) applies to higher

frequencies when 1 and the spectrum follows that of the emission coefficient

(Eq. 8.1) whose frequency response is that of the Gaunt factor, being flat and then

cutting off exponentially at high frequency. In this part of the spectrum, the emission

depends on both the temperature and density. In the flat part of the spectrum (from

Eqs. 8.2 and 8.8, and 8.5) the specific intensity, I / ne2 l=Te

1=2 / EM=Te1=2, so the

source brightness is much more sensitive to density than temperature.

I ν

ν (Hz)

HII Region

Intra-Cluster Gas (x 10^8)

Pla

nck τ << 1

τ >

> 1

1E-16

1E-17

1E-18

1.0E+07 1.0E+09 1.0E+11 1.0E+13 1.0E+15 1.0E+17

Figure 8.3. Two examples of the thermal Bremsstrahlung spectrum with ordinate (I) in cgsunits. For the HII region, the parameters are: ni ¼ ne ¼ 100 cm3, Te ¼ 104 K, l ¼ 5 pc, Z ¼ 1,gff constant ¼ 6:5. For the hot intracluster gas: ni ¼ ne ¼ 104 cm3, Te ¼ 106 K, l ¼ 1Mpc, Z ¼ 1, gff constant ¼ 1:0, and the specific intensity has been multiplied by a factor of108 to appear on the plot. For the H II region, the optically thick and optically thin regions aremarked as well as the Planck curve at an equivalent temperature. The high frequency exponentialcutoffs are evident for both curves but the optically thick self-absorbed region for the intraclustergas occurs at a frequency lower than shown on the plot.


Now that we know the spectrum of free–free emission and how it depends on the

parameters of the source, we can use the observed specific intensity of the source to

obtain the source parameters. From the Bremsstrahlung spectrum alone, this could

be accomplished by making a measurement at low frequency in the optically thick

regime to obtain Te and then making another measurement in the optically thin

regime to obtain ne. Note that the line-of-sight distance through the source must also

be known, but this is often assumed to be about equal to the source diameter, the latter

obtained from the source distance and apparent diameter (Eq. (B.1)). Once the density

is found, the hydrogen mass, MHII, or total gas mass, Mg of the ionized region can be

found from,

MHII ¼ mH ne V ¼ X Mg ð8:9Þ

where mH is the mass of the hydrogen atom, V is the volume of the ionized region, and

X is the mass fraction of hydrogen (X¼ 0:707 for Solar abundance, Eq. 3.4). Note that,

although dust may play an important role in intercepting ionizing photons, it con-

tributes negligibly to the total mass (Sect. 3.4.1).

Free–free emission is most commonly seen in the radio part of the spectrum

when the emission is from gas at 104 K such as HII regions around hot young

stars, and in the X-ray part of the spectrum when the gas is at 1067 K such

as diffuse gas in clusters of galaxies or halos around individual galaxies. As

Figure 8.3 illustrates, the spectrum is broader than these wavebands alone but

free–free emission is most easily isolated from other processes in these wave-

bands. We will therefore elaborate on radio and X-ray waveband observations in

the next sections.

8.2.2 Radio emission from HII and other ionized regions

Regions of ionized gas, such as HII regions (see Figure 8.4), emit by thermal

Bremsstrahlung and properties of these regions can be derived through observations

of their radio emission. Radio observations are exceedingly important in astrophysics

because radio waves are not impeded by interstellar dust. While optical observations

must contend with extinction (Sect. 5.5), radio wavelengths are much longer than

interstellar grain sizes so can pass through without interaction (recall that Rayleigh

scattering / 1=4 when the particle size, Appendix D.3). Thus, measurements in

this waveband can provide information about the source without having to make

often uncertain corrections for extinction. In some cases, especially for HII regions

that are embedded in dusty molecular clouds or behind many magnitudes of extinction

in the plane of the Milky Way, the region cannot be seen at all optically and observations

at radio wavelengths may be the only way of detecting them (see, for example,

Figure 9.6).

For any HII region at radio wavelengths, h k Te (Prob. 8.4) and certain

simplifications can be made to the thermal Bremsstrahlung equations. The Planck

curve, for example, can be represented by the Rayleigh–Jeans approximation


(Eq. (4.6)), so the optically thick emission at low frequency can be represented by in

(cgs. units),

I ¼ BðTeÞ ¼ 2 2

c2k Te / 2 ð 1Þ ð8:10Þ

For optically thin emission, we require a value for the Gaunt factor. When

Te < 9 105 K, which will be the case for HII regions, the Gaunt factor can be

represented analytically by (Ref. [96]),

gffð;TeÞ ¼ 11:962 Te0:15 0:1 ð8:11Þ

DE

CL

INA

TIO

N (

J200

0)

RIGHT ASCENSION (J2000)

21h54m00s 53m45s 30s 15s 00s 52m45s

47°22¢

20¢

18¢

16¢

14¢

12¢

10¢

Figure 8.4. The HII region, IC 5146, at a distance of D ¼ 0:96 kpc. The greyscale image is anR-band optical image. (Reproduced courtesy of the Palomar Observatory-STSc I Digital Sky Survey,Cal Tech). The contour overlays have been made from 92 cm data of the Westerbork Northern SkySurvey (WENSS, Ref. [134]). Contour levels are at 5, 10, 20, 40, 55, 70, 159, and 300 mJy beam1

and the map peak is 318 mJy beam1. The beam is Gaussian with a FWHM of 8000. The total radioflux at this wavelength is S92 cm ¼ 1:44 Jy and the electron temperature is Te ¼ 9000 K. Theionizing star, BDþ463474, can be seen at the centre. The strong point source at the top of theimage is an unrelated background source.


Since the frequency dependence of the optically thin spectrum follows the frequency

dependence of the Gaunt factor (Eq. (8.2)), Eq. (8.11) explicitly shows the flat nature of

the optically thin spectrum. With these assumptions and expanding the term in brackets

in Eq. (8.4), the latter expression for optical depth can be rewritten in common

astronomical units as,

¼ 8:24 102 Te

K

1:35

GHz

h i2:1 EMpc cm6

ð8:12Þ

The brightness temperature of an optically thin source can now be written by using

Eq. (8.12) with Eq. (6.33) to find,

TB ¼ 8:24 102 Te

K

0:35

GHz

h i2:1 EMpc cm6

ð 1Þ ð8:13Þ

Using Eq. (8.13) with the Rayleigh–Jeans relation (Eq. 4.6) again, we can see the

frequency dependence of the specific intensity,

I ¼ 2 2

c2k TB / 0:1 ð 1Þ ð8:14Þ

It is sometimes more useful to measure the total flux density, f , of an HII region which

is related to the specific intensity via f ¼ I (Eq.(1.13)). Together with the Ray-

leigh–Jeans equation, the brightness temperature can then be expressed in terms of

source flux density and Eq. (8.13) can be re-arranged to solve, explicitly, for various

parameters of an HII region. These are the electron density, ne, the hydrogen mass,

MHII, the emission measure, EM (defined in Eq. 8.5), and the excitation parameter,

U (see Eq. (5.4)). For the specific geometry of a spherical HII region of uniform density,

the results are (Ref. [151]),

ne

cm3

h i¼ 175:1

GHz

h i0:05 Te

K

0:175f Jy

0:5D

kpc

0:5 sph

arcmin

1:5

ð8:15Þ

MH II

M

¼ 0:05579

GHz

h i0:05 Te

K

0:175f Jy

0:5D

kpc

2:5 sph

arcmin

1:5

ð8:16Þ

EMpc cm6

¼ 8920

GHz

h i0:1 Te

K

0:35f Jy

sph

arcmin

2

ð8:17Þ

Upc cm2

¼ 4:553

GHz

h i0:1 Te

K

0:35f Jy

D

kpc

2( )1=3

ð8:18Þ


where7 sph is the angular diameter of a spherical HII region as seen on the sky and D is

its distance. A measurement of flux density at a radio frequency in the optically thin

limit, with a knowledge of the temperature and distance, are sufficient to obtain these

quantities (Prob. 8.5). Table 3.1 provides some typical results. The excitation para-

meter (see Eq. 5.4) can then be related to the properties of the ionizing star or stars, as

was described in Example 5.4. Although many HII regions are not perfectly spherical,

their line-of sight distances are not usually significantly different from their diameters,

leading to only minor adjustments in the above equations.

Historically, this approach has most often been applied to HII regions but, in

principle, the above equations for the radio regime apply to any discrete ionized

region, provided Te 9 105 K.

Planetary nebulae, for example, which are ionized gaseous regions formed by the

ejecta of dying stars (Sect. 3.3.2), have properties that are not very different from HII

regions. They are ionized by a central white dwarf and have similar temperatures with

(on average) higher densities. Radio observations of free–free emission from these

nebulae have also been carried out, revealing their properties (see Table 3.1). Planetary

nebulae have historically been a more challenging target than HII regions, however,

because of their comparatively smaller sizes (less than a pc).

Another example is ionized gas in the Milky Way that is not in discrete regions, but

rather spread out throughout the ISM. This is diffuse, low density (e.g. 0:01 cm3

though it varies) gas and therefore the emission from this ISM component is weak. It is

called either thewarm ionizedmedium (WIM, more often used for the Milky Way) or the

diffuse ionized gas (DIG, more often used when observed in other galaxies). The WIM/

DIG is likely caused by ionization of interstellar HI by hot stars that are close to the

boundaries of their parent clouds so that ionizing photons leak out of the surrounding

HII regions into the general ISM. The above equations would require a modification for

geometry if the line of sight through the gas is significantly different from the diameter,

but otherwise are still applicable. The challenge in extracting information about this

component in the radio regime, however, is that the free–free radio emission tends to be

contaminated by synchrotron emission (see Sect. 8.5), requiring that the synchrotron

fraction, if known, be subtracted first. Other methods for probing this diffuse component

are also available, fortunately, such as observing the signals from background pulsars in

the case of the Milky Way (Sect. 5.3) or measuring optical line emission (Sect. 9.4.2).

8.2.3 X-ray emission from hot diffuse gas

The X-ray wave band between about 0:1 keV and 5 keV (the ‘soft’ X-ray band, corre-

sponding to 1016 ! 1018 Hz) is dominated by emission from hot (1068 K),

7The equation for electron density, ne, includes contributions from singly ionized He, if present, so the hydrogenmass, MHII, computed from Eq. (8.16) should be reduced by a correction factor, 1=½1 þ NðHeIIÞ=NðHIIÞ, toaccount for this, where NðHeIIÞ=NðHIIÞ is the ratio of abundances of singly ionized He to ionized hydrogen. ForSolar abundance, this ratio should be less than 8.5 per cent, depending on how many He atoms are ionized, andtherefore we have left the correction factor out of Eq. (8.16).


low density (104 to 102 cm3) gas that emits via thermal Bremsstrahlung. In this

band, the gas is always optically thin (Prob. 8.6). Since h n k Te, the exponential

term in Eqs. (8.1) and (8.2) is not approximately 1 as it was for radio observations of

HII regions. At 1 keV, for example, the gas temperature would have to be higher than

108 K for h k Te. Therefore the exponential term in these equations is important

and the emission is observed on or near the high frequency exponential tail (Figure

8.3). This also means that the electron temperature can be determined (Sect. 8.2.1) by a

fit to the shape of the spectrum in this region.

A challenge of observing in the soft X-ray band is that these photons are very easily

absorbed in the neutral hydrogen clouds that are abundant in the ISM8 . This is beautifully

illustrated by Figure 8.5 which shows the entire sky in the soft X-ray band and in HI. The

Figure 8.5. Illustration of the anti-correlation between the soft X-ray emission and total HIdistribution in the Galaxy, shown over the whole sky in this Hammer–Aitoff projection (see Figure3.2). The centre of the map is the direction towards the nucleus of the Milky Way, called theGalactic Center (GC), and the right and left edges correspond to the anti-centre direction, 180 fromthe GC. The top and bottom of the maps correspond to the North Galactic Pole (NGC) and SouthGalactic Pole (SGP), respectively, and the plane of the Milky Way runs horizontally through thecentre. See Figure 3.3 for a map of the Galaxy as it would be seen from the outside. (a) The 0.25 keVsoft X-ray band, observed with the ROSAT satellite (Ref. [158]). The long curved streaks areartifacts. Most of the emission is seen towards the Galactic poles because much of the X-rayemission along the plane of the Galaxy has been absorbed in neutral hydrogen clouds. (b) All-skymap of the total (summed over all velocities) neutral hydrogen emission in the Milky Way (Ref.[46]). Most of the emission is along the Galactic Plane.

8The absorption is mainly by bound electrons in the heavy elements (e.g. oxygen) that are within these HI clouds.


X-ray map is dominated by free–free emission from hot gas, though contributions from

other emission processes are also present. Most of the X-ray emission is seen in the

direction of the Galactic poles because, along the Galactic plane, X-rays are absorbed in

the abundant HI clouds of the disk of the Galaxy (see Figure 3.3 for a sketch of the Milky

Way). A correction for this absorption must first be applied, therefore, before an accurate

X-ray intensity can be found. Since we have a good map of HI for our Galaxy, such a

correction is routinely carried out, though it adds to the error bar of the result.

Once a correction for interstellar absorption has been made, it is common to quote an

X-ray luminosity (in erg s1) for the object. This can be obtained by converting the

specific intensity (Eq. 8.8) to a luminosity (Prob. 8.7). Although thermal Bremsstrah-

lung emission is expected to dominate in the soft X-ray band, some corrections may

still be required to account for contributions to the luminosity from emission mechan-

isms other than thermal Bremsstrahlung alone. Possibilities include free–bound emis-

sion (Sect. 8.3) or line emission. Assuming that the emission represents only the

free–free component, the luminosity of a uniform density, constant temperature plasma

with Z ¼ 1 and ne ni is,

LX ¼ 6:84 1038 ne2

Te0:5

V

Z 2

1

gffð;TeÞ eh k Te d ð8:19Þ

where 1, 2 are the lower and upper frequencies of the band in which the emission is

observed and V is the volume occupied by the emitting gas.

Taking the Gaunt factor to be a constant (gffð;TeÞ gX) and using this equation

and Eq. (8.9), the electron density and mass of the ionized hydrogen gas9 can be found,

ne

cm3

h i¼ ð1:55 1019 fXÞ

LX

erg s1

0:5V

kpc3

0:5Te

K

0:25

ð8:20Þ

MH II

M

¼ ð3:81 1012 fXÞ

LX

erg s1

0:5V

kpc3

0:5Te

K

0:25

ð8:21Þ

where fX is a unitless function given by,

fX ¼ gX ðeE1kTe e

E2kTeÞ

h i0:5

ð8:22Þ

E1 ¼ h 1 and E2 ¼ h 2 being the lower and upper energies of the band over which

the Bremsstrahlung spectrum is observed, respectively. It is common to express the

temperature, not in Kelvins, but rather in energy units (keV) for a more straightforward

comparison to the observational energy band. For example, a temperature of 107 K

corresponds to k Te ¼ 1:38 109erg ¼ 0:863 keV.

The interstellar absorption illustrated by Figure 8.5 is usually considered to be an

undesirable effect that weakens the emission that we wish to observe and introduces

9Similar comments as given in Footnote 7 of this chapter apply here. However, the error may be larger, dependingon metallicity, since at higher temperatures, more electrons may be released from heavy metals.


error into the result. Surprisingly, however, absorption due to HI clouds can actually

benefit our understanding of the X-ray sky. The velocity of any object that emits a

spectral line can be determined via the Doppler shift of the line (Sect. 7.2.1). For HI

clouds in the Milky Way, these velocities can be translated into distances by adopting a

model for the rotation of the Galaxy (Sect. 7.2.1.2). Since HI emits by the 21 cm line

(Sect. 3.4.5, Appendix C) it is possible to place constraints on the distances to various

HI clouds in the Galaxy. This is not true of continuum emission for which we cannot

obtain velocities (e.g. Prob. 7.2). When continuum emission is observed, it could be

coming from anywhere along a line of sight – from nearby hot gas or from the distant

Universe. However, a detailed comparison of maps, like those in Figure 8.5, for which

HI cloud distances are known, can help to determine how much X-ray emission is

coming from gas in the foreground or background. If an HI cloud is obscuring the

X-ray emission, for example, then the X-ray continuum must originate from behind the

cloud, placing a limit on the distance to the X-ray-emitting gas. This effect is called

shadowing and it has been used to help us understand the hot gas distribution in the

Solar neighbourhood.

The results of this kind of analysis are not perfect, given the placements of clouds and

the fact that sometimes uncertain corrections are required for X-ray emission from our

own Sun. However, together with other information and analysis, a picture has emerged

that the Sun is located in a Local Hot Bubble (LHB) in the Milky Way, as illustrated by

Figure 8.6. The bubble is irregular in shape but appears to be elongated in the vertical

Figure 8.6. Diagram of the Local Hot Bubble (LHB) within which the Solar System is located. Thetop left image shows the region of the Galaxy that has been blown up for the bottom image. TheLHB, of temperature, 106 K, and density, 103 cm3, is the irregularly shaped white region aroundthe Sun extending of order 50–200 pc in size. Contours and shading are related to density. Variousfeatures are shown, including the chimney axis, which is the approximate direction that hot gasmight be venting into the halo (Ref. [95]). (Reproduced by permission of Barry Welsh)


direction, roughly perpendicular to the plane. Such vertical low density regions are seen

elsewhere in the Milky Way and also in other galaxies and are called chimneys. As the

name implies, these are conduits or tunnels in cooler, denser ISM gas through which hot

gas may flow into the Galaxy’s halo10. They are likely produced by supernovae and stellar

winds whose collective outflows have merged into larger structures.

In a number of spiral galaxies, the collective effects of the venting of hot gas have

produced an observable soft X-ray halo around the galaxy. These are typically

observed in spirals that are edge-on or close to edge-on to the line of sight so that

the halo can be clearly seen independent of the disk. They also support a picture in

which the disk–halo interface of a galaxy is a dynamic place through which outflows

occur, and possibly inflows as well if cooler gas descends again, like a fountain.

There may, however, be other reasons for the presence of X-ray halos around

galaxies. The galaxy, NGC 5746, for example does not have sufficient supernova

activity in its disk to produce the X-ray halo that is observed around it (see

Figure 8.7). It may be that hot gas around this galaxy has been left over from the

galaxy formation process itself (Ref. [119]). Example 8.1 provides an estimate of the

density and mass of this halo.

Example 8.1

Assuming that the soft X-ray halo of NGC 5746 (Figure 8.7) is entirely due to thermalBremsstrahlung, estimate the electron density and hydrogen mass of this halo.

From the figure caption, Te ¼ 6:5 106 K and LX ¼ 4:4 1039 erg s1. The radius

of the emission is about 20 kpc leading to a volume of V 3:35 104 kpc3,

assuming spherical geometry. The lower and upper bounds of the observing band

are E1 ¼ 0:3 and E2 ¼ 2 keV, respectively. The mid-point of this band (E ¼ 1:15

keV) corresponds (via E ¼ h ) to ¼ 2:8 1017 Hz. From Figure 8.2, the Gaunt

factor is then gX 0:7. By Eq. 8.22, this gives fX ¼ 1:6. Putting the above values into

Eqs. (8.20) and (8.21) yields, ne ¼ 0:002 cm3 and MHII ¼ 1 109M. These

results are only approximate because this halo is likely to have a significantly varying

density with radius. The assumption of pure thermal Bremsstrahlung must also be

considered carefully (see Sects. 8.3 and 8.4).

In clusters of galaxies, immense reservoirs of hot gas have been detected between

the galaxies, visible via their X-ray emission. An example is the galaxy cluster,

Abell 2256, shown in Figure 8.8. This intracluster gas is the dominant baryonic

component of mass in the cluster. In Abell 2256, for example, the total mass

10It is not always clear whether hot gas is currently flowing into the halo through individual chimneys or whetherthey are simply structures that may be relics or result from other factors such as magnetic fields. Therefore, thecooler ‘walls’ of such features that extend above the plane have also been called worms which has no implicationfor hot gas outflow.


contained in the intracluster gas is 18 times greater than the total mass contained in

all the stars of all the galaxies in the cluster (Ref. [141])! Most of this gas is likely a

remnant of the cluster formation process, although a small fraction may have come

from supernova outflows from the galaxies. The latter contribution accounts for a

small (less than Solar) heavy metal enrichment (Sect. 3.3.2). The total mass of the

cluster is even greater than that of the gas, however, consisting of both light and

dark matter components. The total (light þ dark) mass of the cluster may be found

via gravitational lensing (Sect. 7.3.1), via an analysis of the motions of the

individual galaxies (Sect. 7.2.1.2) or by an assumption of hydrostatic equilibrium

for the gas. The latter means that the gas, as a whole, is gravitationally held in place

and not ‘evaporating’ away. With this assumption, it is possible to calculate how

much mass is required to hold gas of a given temperature in place. The results of

such analyses indicate that the gas mass to total mass fraction in clusters of galaxies

is 1520 per cent (e.g. Prob. 8.8). This is in agreement with the baryon fraction

expected from Big Bang nucleosynthesis (Sect. 3.2), suggesting that there may be

Figure 8.7. The galaxy, NGC 5746, is a typical spiral in the optical, displaying a flat disk cut by adust lane. Surrounding this galaxy, however, is a halo of X-ray emission, shown in blue. The X-rayemission is due to thermal Bremsstrahlung from hot gas at Te ¼ 6:5 106 K that extends to about20 kpc from the disk. The X-ray luminosity is LX ¼ 4:4 1039 erg s1 in the 0.3 – 2 keV waveband(Ref. [119]). (Reproduced by permission. X-ray: NASA/CXC/U. Copenhagen/K.Pedersen et al;Optical: Palomar DSS) (See colour plate)


no ‘missing baryons’ in rich clusters of galaxies that have X-ray emitting gas. The

problem of missing non-baryonic matter is, however, still present.

8.3 Free–bound (recombination) emission

The free–bound process, or recombination, has already been discussed in the context of

the ionization equilibrium that is present in HII regions (Sect. 3.4.7 and Example 5.4)

in which the ionization rate must balance the recombination rate. We now wish to

consider recombination as an emission mechanism.

Recombination involves the capture of a free electron by a nucleus into a quantized

bound state of the atom. Therefore, free–bound radiation must only occur in ionized

gases. This now introduces the possibility that both free–bound and free–free radiation

may occur from the same ionized gas, complicating the analysis of such regions. The

free–bound process is essentially the same as described for free–free emission except

DE

CL

INA

TIO

N (

J200

0)


17 08 07 06 05 04 03 02 01 00

78 50

45

40

35

30

25

Abell 2256

Figure 8.8. The cluster of galaxies, Abell 2256 (redshift, z ¼ 0:0581), is shown in the optical(negative grescale) and in X-rays in the observing band, 0.1–2.4 keV (contours). The gas is emittingat k Te ¼ 7:5 keV and the X-ray luminosity is L0:1!2:4 ¼ 3:6 1044 erg s1 (Ref. [56]), of which78 per cent is due to thermal Bremsstrahlung (Ref. [22]). The cluster radius is R ¼ 2:0 Mpc (Ref.[141]). The total mass of the cluster, including all light and dark matter, is M ¼ 1:2 1015 M(Ref. [130]). All values have been adjusted to H0 ¼ 71 km s1 Mpc1. The optical image is in Rband (Reproduced courtesy of the Palomar Observatory-STSc I Digital Sky Survey, Cal Tech). X-raydata were obtained from the Position Sensitive Proportional Counter (PSPC) of the RontgenSatellite (ROSAT)

8.3 FREE–BOUND (RECOMBINATION) EMISSION 251

for the end state, and in fact the free–bound emission coefficient can be written in the

same fashion as Eq. (8.2) (Ref. [96]),

j ¼ 5:44 1039 Z2

Te1=2

ni ne gfbð;TeÞ eð h

kTeÞ ð8:23Þ

Note that this emission coefficient is identical to that of free–free emission except for

the use of the free–bound Gaunt factor, gfb, so an understanding of the relative

importance of free–bound emission compared to free–free emission reduces to an

understanding of the relative importance of the two Gaunt factors.

Computation of the free–bound Gaunt factor, gfbð; TeÞ, requires sums over the

various possible final states in atoms and taking into account a Maxwellian distribution

of velocities for the intitial states. In the limit, if the final state has a very high quantum

number that approaches the continuum, then gfb ¼ gff . Gaunt factors have been

determined for a range of temperatures, frequencies, and metallicities by various

authors (e.g. Refs. [28], [107]). In Figure 8.2 we show one result, the sum of

gfb þ gff , for the temperature, Te ¼ 1:58 105 K (grey curve). The free–bound

Gaunt factor is therefore represented by the excess over the smooth gff curve below it.

Note that there are sharp edges, corresponding to transitions into specific quantum

levels of energy, En (see Eq. C.5), followed by smoother distributions to the right

(higher frequencies) of the edges. This behaviour can be understood by considering

conservation of energy as electrons are captured, the energy difference going into the

emitted photons (Prob. 8.9).

For low frequencies (i.e. h k Te), Figure 8.2 shows that the total Gaunt factor is

just the free–free value, implying that recombination radiation is negligible in compar-

ison to thermal Bremsstrahlung in such a gas. This is typical of other temperatures

as well, validating our assumption (Sect. 8.2.2) that the radio continuum emission

from HII regions and planetary nebulae is due entirely to thermal Bremsstrahlung. In

this regime, the electron is perturbed only slightly from its path. However, once

h ) k Te, gfb may become significant. For the temperature, 1:58 105 K, plotted

on the graph, this corresponds to ) 3 1015 Hz which is in the UV part of the

spectrum. For HII regions and planetary nebulae with temperatures closer to 104 K

(Table 3.1), the corresponding frequency regime is ) 2 1014 Hz which is in the

near-IR and optical part of the spectrum. Thus, depending on temperature and fre-

quency band, the recombination continuum can be a significant or even dominant

component to the total continuum. As the temperature increases, however, the relative

contribution of gfb decreases. The free–free Gaunt factor has a weak dependence on Te

(see Eq. 8.11 or Figure 8.2), increasing as Te increases. However, gfb has a stronger

dependence on temperature and also decreases as Te increases. At higher temperatures,

the electrons will have higher speeds, on average, and are less likely to be captured.

Therefore at very high temperatures, free–free emission dominates the continuum. A

final important note is that a proper calculation of Gaunt factors requires all elements

to be taken into account. For example, at temperatures below 106 K, recombinations

to C V, C VI, O VII, O VIII and N VII are important at high frequencies.


A real spectrum of a combined free–free and free–bound continuum is shown

for a Solar flare at X-ray wavelengths in Figure 6.7. Several spectral lines are also

visible in that spectrum. Figure 8.9 shows another example, this time for an HII region

at optical wavelengths. In this spectrum, the optical continuum consists of free–bound

radiation, two-photon emission (see next section) and scattered light from dust. In

addition, a great many optical emission lines are present so a measurement of the

continuum must be made at frequencies between the lines. These figures demonstrate

that it is sometimes difficult to isolate emission from a single process alone. They are

examples of complex spectra for which modelling is required to understand the various

contributions. Complex spectra will be discussed further in Sect. 10.1.

8.4 Two-photon emission

Another important contribution to the continuum in low density ionized regions at

some frequencies is a process called two-photon emission. Two-photon emission

Figure 8.9. A spectrum of an HII region in the optical waveband. Many strong emission lines canbe seen superimposed on the underlying continuum. The dashed curve shows the combinedcontinuum from both free–bound radiation and two-photon emission. The remainder of thecontinuum is due to scattered light from dust. The spectrum is of a region in the 30 Doradus nebula,also called the Tarantula Nebula, which is in the Large Magellanic Cloud, a companion galaxy to theMilky Way. (Reproduced by permission of Darbon, S., Perrin, J.-M., and Sivan, J.-P., 1998, A&A,333, 264)

8.4 TWO-PHOTON EMISSION 253

occurs between bound states in an atom but it produces continuum emission instead of

line emission. It occurs when an electron finds itself in a quantum level for which any

downwards transition would violate quantum mechanical selection rules (Appendix C)

and the transition is therefore highly forbidden (e.g. Appendix C.4). The electron

cannot remain indefinitely in an excited state, however, and there is some probability,

although low, that de-excitation will occur. The most likely de-excitation may not be

the usual emission of a single photon, but rather the emission of two photons. These two

photons can take on a range of energies whose sum is the total energy difference

between the levels. Two-photon emission will occur as long as the electron is not

removed from the level via a collisional energy exchange first. Therefore, it occurs in

lower density ionized nebulae in which the time between collisions that would

depopulate the problematic level is longer than the two-photon lifetime.

The hydrogen atom provides a good example of two-photon emission. We have

already seen that the Ly line (a recombination line, Sect. 3.4.7), which results from

a de-excitation of an electron from the n ¼ 2 to n ¼ 1 levels, has a very high

probability (Table C.1). However, the n ¼ 2 quantum level consists of both 2s and

2p states (Table C.2). The transition 2p ! 1s is a permitted transition with

A2p! 1s ¼ 6:27 108 s1, but the 2s ! 1s transition is highly forbidden. The for-

bidden transition will occur via two-photon emission with a rate coefficient of

A2s! 1s ¼ 8:2 s1, considerably lower than the 2p ! 2s transition, but still high

enough for this process to be important in nebulae of densities< 104 cm3 (Ref. [155]).

Since the sum of the energies of the two photons must equal the energy difference

between the two levels, then so must the frequencies (E ¼ h ),

1 þ 2 ¼ Ly ð8:24Þ

where 1 and 2 are the frequencies of the two photons and Ly is the frequency of

Ly. The spectrum of photon frequencies can take on any value as long as Eq. (8.24) is

satisfied, but the most probable configuration is that the two photons have the same

frequency. Therefore, the probability distribution of photon frequencies is symmetric

and centred on the midpoint frequency which, for Ly, is 0;2ph ¼ 1:23 1015 Hz

(243 nm). This is in the ultraviolet part of the spectrum with a width that stretches to

¼ 0 and ¼ Ly, the probability being lower as the photon frequency departs from

0;2ph. Note that since the number of photons with frequencies above 0;2ph is equal to

the number below, the energy distribution (which describes the spectrum) is actually

skewed to higher energies.

The strength of two-photon emission depends on the number of particles in the

n ¼ 2 state of hydrogen. This, in turn, depends on the recombination rate to this

level which is a function of both density and temperature. Therefore, two-photon

emission, although quantum-mechanical in nature, can be thought of as thermal

emission and the density dependence is the same as for free–free and free–bound

emission, i.e. j / ne2. The emission coefficient decreases with increasing tempera-

ture, described by a somewhat more complex function. Further details can be found in

Ref. [96].


As noted in Sect. 8.3, two-photon emission is present in the optical continuum of

the HII region shown in Figure 8.9. It tends to be a smaller contributor to the

spectrum than free–free or free–bound emission except at frequencies near its peak.

For example, at Te ¼ 104 K, two-photon emission is about 30 per cent higher than

the sum of free–bound and free–free emission at 0;2ph ¼ 1:23 1015 Hz. However,

at ¼ 3:0 1014 Hz it is about six times weaker and, once the frequency departs

significantly from 0;2ph, two-photon emission is negligible (Ref. [27]). However,

again, all species must be considered when computing the continuum from an

ionized gas. Two-photon emission is also possible from He as well as heavier

species, for example. In hotter gas, two-photon emission from highly ionized species

such as N VI, Ne X, Mg XI, Si XIV, SX VI, Fe XXV, and others cannot be ignored if

these elements are present. For gas between 106 and 107 K, two-photon emission can

again dominate within certain parts of the observing band between 0:1 and 1 keV

(Ref. [131]).

8.5 Synchrotron (and cyclotron) radiation

Synchrotron radiation results from relativistic electrons (those with speeds approach-

ing the speed of light) moving in a magnetic field. The source of relativistic electrons

is the electron component of the cosmic rays that were discussed in Sect. 3.6.

Synchrotron radiation is widespread in the Milky Way and other galaxies and is

one of the most readily observed continuum emission processes in astrophysics.

There is some similarity between the Bremsstrahlung radiation discussed in Sect. 8.2

and synchrotron radiation. In the former case, the electron is accelerated in an

electric field, ~E, and in the latter case, the electron is accelerated in a magnetic

field, ~B. Both represent ‘scattering’ of a charge in a field. Therefore, synchrotron

emission is sometimes referred to as magnetic Bremsstrahlung or magneto-

Bremsstrahlung radiation.

To understand the emission, it is helpful to recall that a charged particle moving at a

velocity,~v, in a magnetic field,~B, experiences a Lorentz force which, for an electron of

charge, e, and negligible electric field is~Fe ¼ e ~vc ~B

(see Table I.1). Such a force

exists whether or not the electron is relativistic. The direction of the force is perpendi-

cular to both~v and ~B and the magnitude of the force is,

Fe ¼ e v

cB sin ¼ e v

cB? ð8:25Þ

where is the angle between ~v and ~B, called the pitch angle, and B? ¼ B sin .

Figure 8.10 illustrates this geometry. If ~v is perpendicular to ~B (sin ¼ 1), the

electron experiences the maximum force and will circle about the field line. If the

electron moves only parallel to~B (sin ¼ 0), then it experiences no force and will not

radiate. If the initial velocity is in an arbitrary direction, then the electron will spiral

about the field line. Over a period of gyration (the time for a loop), the speed of the

8.5 SYNCHROTRON (AND CYCLOTRON) RADIATION 255

electron does not change due to this force11, but its change of direction constitutes an

acceleration and it will therefore radiate. Note that any charged particle, including

protons and other ions, experience this behaviour although, as we will see, the

synchrotron emission from ions is negligible in comparison to that of electrons.

If the magnetic field within a plasma is strong enough that it dynamically affects

charged particles, it is called a magnetized plasma. A more specific criterion is if a

charged particle completes at least one gyration before it interacts with (collides

with) another particle, a condition that may be different between electrons and ions.

In a magnetized plasma, the gyrating charged particle is ‘coupled’ to the field and

therefore, in a sense, trapped by it. If the field lines curve, for example, the gyrating

particle will follow along this curvature. Any emission that may be given off by such

trapped particles can then be used to map out the magnetic field configuration. A

good example is the prominences seen on the Sun, as shown in Figure 6.6 or the

coronal loop of Figure 8.11. These particular images show recombination line

emission (rather than synchrotron) associated with the ionized gas (Sect. 3.4.7).

Such emission traces out the magnetic fields that loop out of the Sun, forming the

delicate filamentary structures shown. In this way, trapped charges result in the

‘illumination’ of the otherwise invisible magnetic fields. Similarly, synchrotron

emission allows us to map out other cosmic magnetic fields in locations where the

electrons are relativistic.

f B sinf

Instantaneousvelocityvector

Instantaneousvelocityvector

B

e–

q = 1g

q = 1g

(a) (b) (c)

BB

Figure 8.10. Illustration of the radiation given off by an electron with instantaneous velocity, ~v,encircling magnetic field lines of strength, B. (a) Non-relativistic electron whose velocity isperpendicular to the direction of the magnetic field. The force acting on the electron isperpendicular to both ~v and ~B, causing the electron to circle about the field lines. The resultingcyclotron radiation is emitted in all directions. (b) Relativistic electron whose motion isperpendicular to the magnetic field. The force on the electron is in the same direction as in (a) butthe radiation given off (synchrotron radiation) is highly beamed in the forward direction into a coneof angular radius, ¼ 1=, where is the Lorentz factor (see Eq. 3.37). (c) More realisticsituation in which an electron has components of its velocity both parallel and perpendicular to thedirection of B. The pitch angle, , is the angle between ~v and ~B.

11Eventually the loss of energy due to radiation will slow the particle, but this occurs over a timescale that is verylong in comparison to a gyration time.


Radiation that results specifically from motion about the magnetic field will depend

on the strength of the field, B. Therefore, not only do observations of such emission

provide a means of determining the presence and orientation of fields, but they also

provide a means of obtaining the magnetic field strength (e.g. Table 8.1). We will see

how this dependence occurs in the next sections. Before considering the details of

Figure 8.11. A coronal loop can be seen at the edge of the Sun. This image, taken with theTransient Region and Coronal Explorer (TRACE) satellite, shows emission at 171 A from Fe IX andFe X ions that trace out delicate filamentary and loop-like magnetic field lines in the Solaratmosphere. This emission is characteristic of plasmas at Te ¼ 6 105 K in the upper transitionregion of the atmosphere (see Figure 6.5). (Reproduced by permission. TRACE is a mission of theStanford-Lockheed Institute for Space Research, part of the NASA Small Explorer program)

Table 8.1 Sample magnetic field strengthsa

Object B (G)

Interstellar space 106

Interplanetary space 106 105

Solar corona 105 100

Planetary nebulae 104 103

H II region 106

Pulsar (surface) 1012

Supernova remnants (SNRs) 105 102

Earth 0:31

Jupiter 4:28

Saturn 0:22

Uranus 0:23

Neptune 0:14

aRef. [96] except for the planets and SNRs. (Ref. [12]) Planetaryfields are surface equatorial values (but note that the value isvariable with position and time).


synchrotron radiation, however, it is helpful to start with its non-relativistic, and less

frequently observed, counterpart – cyclotron radiation.

8.5.1 Cyclotron radiation – planets to pulsars

When a non-relativistic electron gyrates about the magnetic field, cyclotron radiation is

emitted at the frequency of this gyration, called the gyrofrequency, 0. For circular

motion, the acceleration always points to the centre of the circle and the electric field

vector of the emitted radiation follows that of the acceleration vector as seen by a

distant observer (see Ref. [101], pp. 62–66 for details on the emission of an accelerating

charge). If seen from a direction along the field line,~E will rotate with time, giving rise

to circular polarization of the emission. Seen from the side, the electric field vector will

oscillate linearly (like a dipole), giving rise to linear polarization. From an intermediate

angle, the polarization is elliptical. A characteristic of this kind of emission, therefore,

is that it is intrinsically polarized and high degrees of polarization (Dp, see Sect. 1.7)

have been measured in sources in which cyclotron radiation has been observed.

Examples are the planets that have magnetic fields – the Earth, Jupiter, Saturn, Uranus,

and Neptune – for which Dp up to 100 per cent has been measured (Ref. [187]).

For spiral motion, the electron has both a velocity component that is parallel to the

magnetic field and therefore unaffected by it, vk, as well as a perpendular component,

v? ¼ v sin . The Lorentz force (Eq. 8.25) can then be equated to the centripetal force,

Fe ¼ e v

cB? ¼ me

v?2

r0

ð8:26Þ

where r0 is the orbital radius that is perpendicular to the field, called the radius of

gyration or gyroradius. For protons or other charged particles, the appropriate mass and

charge must be used (Prob 8.10). Eq. (8.26) can be solved for r0,

r0 ¼ me v? c

e B¼ v? T

2¼ v?

2 0

ð8:27Þ

where T ¼ 1=0 is the period of gyration and we have used the relation between

velocity, distance and time (v? ¼ 2 r0 =T ). The gyrofrequency then follows from

Eq. (8.27),

0 ¼ e B

2me c¼>

0

MHz

h i¼ 2:8

B

Gauss

ð8:28Þ

Note that the electron gyrofrequency is independent of velocity and therefore inde-

pendent of the kinetic energy of the electron (Ek ¼ 12

me v2) when v c. Even if

there is a range of particle velocities, as is normally the case, all particles will emit at

the same frequency for the same magnetic field strength, though the radius of gyration

will be different. Since the gyrofrequency is single-valued, it really represents


monochromatic, or line radiation (see Chapter 9), rather than continuum radiation, with

a width that can depend on various line widening mechanisms (e.g. Sect. 9.3).

However, the physical process is fundamentally different from that of ‘normal’ spectral

lines which result from quantum transitions within atoms and molecules. Example 8.2

provides a numerical example.

Example 8.2

Determine the radius of gyration, period and gyrofrequency of a typical electron within thewarm ionized medium (WIM) of the Milky Way, in which the magnetic field strength isB 3 mG. Can we expect to observe the resulting cyclotron radiation?

From Sect. 8.2.2, the WIM consists of ionized gas with Te 104 K and

ne 0:01 cm3. From Eq. (8.28) with the given magnetic field strength, we find,

0 ¼ 8:4 Hz so T ¼ 1=0 ¼ 0:12 s. A ‘typical’ electron might be expected to have

the mean electron speed in a Maxwellian velocity distribution which, for a gas at 104 K,

is v ¼ 1:1 108 cm s1 (see Eq. 3.9). Therefore, letting v v? and using Eq. (8.27),

the radius of gyration is r0 ¼ 2:1 106 cm or 21 km. A gyrofrequency of 8:4 Hz is

extremely low and the emission could not be observed from the ground because it would

not propagate through the Earth’s ionosphere (see Figure 2.2). Moreover, the plasma

frequency in the WIM is (see Eq. E.9) p ¼ 8:9 n1=2e ¼ 0:89 kHz, more than 100

higher than the cyclotron frequency, so this radiation would not even escape from the

WIM. Therefore, this emission cannot be observed.

The importance of Eq. (8.28) is that the magnetic field strength can be immediately

obtained if the cyclotron frequency can be measured. The problem, however, is that this

frequency is very low, as demonstrated by Example 8.2. The cyclotron frequency must

be higher than the plasma frequency for the radiation to escape, favouring objects with

high magnetic fields (see Table 8.1) and/or low electron densities (Prob. 8.10). Also, the

magnetic field must be higher than about 3:5 G in order for this emission to be

detectable from the ground because of the Earth’s 10 MHz ionospheric cutoff

(Figure 2.2). Thus, the number of objects for which cyclotron emission has so far

been observed is limited to those with strong fields, or those for which satellite or space

probe data are available, or both. Examples are pulsars, the Sun, and the planets that

have magnetic fields (see Table 8.1).

For low velocity electrons (v9 0:03 c) only a single gyrofrequency at 0 is observed,

as we have seen (Ref. [14]). In a thermal distribution of particles, this corresponds

to Te 9 106 K. In hotter gases or in non-thermal distributions with higher velocities,

however, not only is the fundamental frequency, 0, observed, but also weaker

harmonics at integer multiples of the fundamental12. The observation of multiple

‘spectral lines’ at the fundamental frequency and its harmonics helps to confirm that



the emission is actually due to the cyclotron process. Such lines have been observed in

emission in some regions and absorption in others, depending on the presence and

strength of the background radiation field. Example 8.3 provides an example.

Example 8.3

Three absorption lines have been observed in the X-ray spectrum of a pulsar, named1E1207.4-5209, at energies of 0:7, 1:4 and 2.1 keV (Ref. [21]). If the gravitational redshiftfrom this neutron star is zG ¼ 0:2, what is the strength of its magnetic field?

These three lines are equally spaced in frequency and are presumed to represent the

cyclotron fundamental frequency and its first and second harmonics. The fundamental

gyrofrequency is then (via E ¼ h ) 0 ¼ 1:69 1017 Hz. For a gravitational red-

shift of zG ¼ 0:2, the true frequency (Eq. 7.1 with ¼ c) is 0 ¼ ð1 þ zGÞ ¼2:03 1017 Hz. Then using Eq. (8.28), B ¼ 7:1 1010G.

Other cyclotron spectra may not be quite as straightforward to interpret as is implied

by the neutron star of Example 8.3. For example, the magnetic field strength may vary

significantly within the region being observed, resulting in a number of fundamental

cyclotron frequencies, each corresponding to a different field strength. For a smoothly

varying field, the result will be emission that is spread out over a corresponding

frequency range, giving continuum, rather than line emission. This can be seen in

Figure 8.12 which shows the low frequency radio spectra of the planets.

Low frequency planetary radio emission comes from a region around the planets

called the magnetosphere which is the region within which charged particles are more

strongly affected by the magnetic field of the planet than by other external fields. A

diagram of Jupiter’s magnetosphere is shown in Figure 8.13. This topology, that of a

magnetic dipole swept by the Solar wind (Sect. 3.6.3), is similar for the other planets

that have magnetic fields. Jupiter’s field is both larger and stronger than those of the

other planets and it also has an additional component related to its Galilean13 satellite,

Io, whose volcanic outbursts supply additional plasma into a torus, called the Io plasma

torus, around the planet.

The magnetic field strength varies with position within the magnetosphere. The

strongest magnetic fields are near the poles of the planet, as indicated by a higher

density of field lines in Figure 8.13. From Figure 8.12, Jupiter’s maximum gyrofre-

quency is 0 40 MHz which, by Eq. (8.28), gives a maximum magnetic field

strength of B 14 G, corresponding to a location approximately where the magnetic

field lines that pass through the Io plasma torus intersect the planet (Ref. [187]).

However, its equatorial surface field is B 4 G (Table 8.1). The radio emission also

varies with time as the planet rotates, as the various satellites orbit through the

magnetosphere, as new material is injected into the magnetosphere, and as the Solar

13The four largest moons of Jupiter, Io, Europa, Ganymede, and Callisto, are called the Galilean satellites, aftertheir discoverer.


wind varies. The result is strong intermittent radio bursts. Depending on the type of

stellar system, similar radio bursts might be detectable from planets orbiting other stars

as well (Prob. 8.11), providing a possible additional probe of extrasolar planets.

An accurate description of the radio spectra shown in Figure 8.12 involves complex

modelling that takes a variety of effects into account. For example, for particles that are

more energetic or mildly relativistic, the cyclotron frequencies and line widths are

functions of electron energy. Therefore, the shape of the spectrum depends on the

electron energy/velocity distribution which tends not to be Maxwellian (i.e. it is not

thermal). At the low frequencies shown for the planets, the electron distribution is

actually inverted. Rather than the declining power law distribution that we saw for

cosmic ray nucleons (Figure 3.25) and electrons (Eq. 3.43), there are instead more high

energy electrons than low energy ones. This is because electrons with high pitch angles

(see Figure 8.10c), which tend to be those of higher energy, are more easily reflected

back along the field lines when they enter the ‘bottleneck’ region of high magnetic field

strength and high plasma densities near planetary poles. Electrons with smaller pitch

angles are more likely to suffer collisions in these regions and are not reflected.

Therefore, the low energy particles are selectively removed. Other effects, such as

resonances between the electron gyrofrequency and ambient radiation frequencies from

other emitting electrons are also important and can result in very intense, low frequency

NEPTUNE URANUS

Frequency (kHz) 100 1000 10000

Flu

x D

ensi

ty a

t 1

A.U

. (W

m–2

Hz–1

)

10–1

9

10

–21

10–2

3

EARTH

SATURN

JUPITER

(S-bursts)

(lo)

(non-lo)

Figure 8.12. Low frequency radio spectra of the planets that have magnetic fields. The verticaldashed line marks approximately the Earth’s ionospheric cutoff. Jupiter has the strongest emission,having a higher magnetic field (Table 8.1) and a larger magnetospheric cross-section for interactionwith Solar wind particles. The interaction of Jupiter’s magnetosphere with its inner Galilean moon,Io, is responsible for higher frequency emission from this planet as well as the ‘S-bursts’ which referto short duration radio bursts. During radio bursts, peak flux densities can be >10 higher thanshown here and, for Jupiter, >100 higher. The flux density scale assumes that the radio emissionhas been measured at a distance of 1 AU from each planet (Adapted from Ref. [187]).


radio emission14. Scattering of electrons from waves in the magnetic field lines (called

Alfven waves), plasma waves, and other processes can also affect the shape of the

spectrum. Synchrotron radiation is also produced in the magnetosphere of Jupiter.

Those particles that do leak through to the upper atmosphere in the regions of the

magnetic poles can collide with and ionize atmospheric atoms and molecules. When

the electrons recombine, the subsequent downwards-cascading bound–bound transi-

tions (i.e. fluorescence, see Sect. 5.1.2.1) can produce visible light. For the Earth, the

result is the aurora borealis (northern lights) in the northern hemisphere and the aurora

australis (southern lights) in the southern hemisphere. These are most often seen after a

Solar flare (Sect. 6.4.2), demonstrating the sensitive interaction between the Earth’s

magnetosphere and the Solar wind.

8.5.2 The synchrotron spectrum

There are a number of important ways in which the synchrotron spectrum differs from

that of cyclotron emission. As we did for thermal Bremsstrahlung radiation (Sect. 8.2),

Figure 8.13. Diagram of Jupiter’s magnetosphere. The planet, itself, is the small dot near thecentre. The curves with arrows show the field lines which come closer together (stronger fields) nearthe poles of the planet. Note the Io plasma torus as well as the asymmetric shape of the field due toits interaction with the Solar wind. (Adapted from http://pluto.jhuapl.edu/science/jupiterScience/magnetosphere.html)

14These effects produce what is called the electron cyclotron maser.


we will describe ‘semi-qualitatively’ the factors that are involved in deriving the

spectrum. More detailed descriptions can be found in Refs. [138], [143], [14], and others.

Firstly, the mass of a relativistic particle is greater than its rest mass by a factor, ,

where is the Lorentz factor given by Eq. (3.37). The energy of a relativistic electron is

therefore, E ¼ me c2 Eq.(3.36). The relativistic gyroradius and relativistic gyrofre-

quency must take this into account, so using Eqs. (8.27) and (3.36) and setting v? c

(note, all in cgs units),

r ¼ r0 ¼ me c2

e B¼ E

eB¼ 2:2 109 E

Bð8:29Þ

and using Eqs (8.29) with (3.36),

¼ 0

¼ e B

2 me c¼ 2:3

B

Eð8:30Þ

and the period, T ¼ 1=. Eqs. (8.29) and (8.30) reduce to Eqs. (8.27) and (8.28),

respectively when the electron is non-relativistic (i.e. 1). Note that these equations

now do depend on electron energy, unlike their cyclotron counterparts. Since Lorentz

factors can be extremely high (Sect. 3.6), the above two equations indicate that the

relativistic gyroradius is much higher than the cyclotron radius and the relativistic

gyrofrequency is much lower than the cyclotron gyrofrequency. At first glance, a very

low value of might suggest that synchrotron-emitting frequencies would be too low to

measure. However, this is not the case, as we will see below. Although Lorentz factors

up to ¼ 1011 have been measured for nucleons, there are very few particles with such

high values (see Figure 3.25) and a ‘typical’ electron energy is lower (Example 8.4).

Example 8.4

Determine the gyroradius, the gyrofrequency, and the period of an electron with ¼ 104 inthe ISM. What is the speed of the electron compared to the speed of light?

For ¼ 104, the electron energy is (Eq. (3.36)) E ¼ me c2 ¼ 8:18 103 erg (5.1

GeV). For a typical ISM magnetic field of 3mG, we find (Eq. 8.29) r ¼ 6:0 1012 cm,

or approximately 86 the radius of the Sun. The gyrofrequency (Eq.(8.30)) is

¼ 8:4 104 Hz, so its period is T ¼ 20 min. By Eq. (3.37), v=c ¼ 0:999999995.

The second important difference with cyclotron radiation is that, because of the

relativistic motion of the electron, the transformations that need to be made between

the rest frame of the electron (the frame moving with vk of the electron) and the rest

frame of the distant observer (the ‘lab frame’) are significant. These transformations15

have a number of consequences.

15The required transformations, called Lorentz transformations for velocity and time, are extended to also includea transformation for acceleration.


For example, in the rest frame of the electron, the power emitted is given by the

Larmor formula (Table I.1) which involves the electron’s acceleration. A transforma-

tion of this acceleration to the lab frame, however, results in a total power that is

boosted by a factor, 2. Moreover, the distribution of this power with angle also

changes. In the electron’s rest frame, the emission is over a very wide angle (see

Figure 8.10a). However, a distant observer will see the emission beamed into a narrow

cone (Figure 8.10b). The opening angle of the cone depends on the particle’s energy

such that higher energy particles (higher ) have narrower cones, the angular radius of

the cone (see Figure 8.10c) being,

¼ 1

ð8:31Þ

The direction of the beam is the direction of the instantaneous velocity vector of the

particle, so as the particle spirals about the field, the cone direction rotates with it and a

distant observer will only see emission when (and if) the cone sweeps by his line of

sight. The opening angle is very small for a highly relativistic electron (Example 8.5)

so the time over which this cone sweeps by the observer is small in comparison to the

larger gyration period, resulting in brief, repeated pulses of emission.

To obtain the synchrotron spectrum of a relativistic gyrating electron requires a

conversion from the time to the frequency domain as well as the final important

departure from the cyclotron case – the need for a Doppler shift (Sect. 7.2.1) since the

observer is only seeing emission when the particle is moving towards him. As before,

the time/frequency conversion requires a more detailed analysis16. As we saw for

thermal Bremsstrahlung radiation (Sect. 8.2.1), however, the shortest time period (the

pulse duration) sets the maximum frequency as seen by the observer. This maximum,

called the critical frequency, above which there is negligible emission, is given by,

crit ¼ 3

22 0 sin ¼ 3 e

4me c2 B? ¼>

crit

MHz

h i¼ 4:2 2 B?

Gauss

ð8:32Þ

where we have used Eq. (8.28). Thus, the maximum frequency is, in fact, very high (see

Example 8.5) and much higher than the relativistic gyrofrequency. Most of the energy,

however, is actually emitted at a frequency somewhat lower than the critical frequency,

i.e. at max ¼ 0:29 crit. The longest time period, which is related to the gyration

period, determines the fundamental frequency, f . The spectrum will contain f and all

its harmonics, where,

f ¼ 1

0

sin2 ¼>

fMHz

h i¼ 2:8

sin2

B

Gauss

ð8:33Þ



using Eq. (8.28) again. The fundamental frequency is very low and its harmonics are so

closely spaced (Example 8.5) that the synchrotron spectrum, unlike the cyclotron

spectrun, is essentially continuous. The end result is a very broad-band continuous

spectrum that peaks at max and becomes negligible at > crit. Since the spectrum

spans frequencies that are typically much greater than the plasma frequency, even small

magnetic fields produce observable synchrotron radiation if a supply of relativistic

electrons is present. It can be shown that the power emitted by a single relativistic

particle is a strong function of the particle rest mass, i.e. P / 1=m4. Therefore, the

emission from relativistic protons, which are more abundant than electrons (see Sect.

3.6.1) is negligible.

Example 8.5

For the same conditions as given in Example 8.4, and assuming that the relativistic electronhas pitch angle of ¼ 45, determine the total opening angle of the emission cone,the critical frequency, and the spacing of the harmonics for this particle.

For ¼ 104 (Eq. 8.31), 2 ¼ 2= ¼ 2 104 rad or 0.01. From Eq. (8.32),

crit ¼ 890 MHz. The harmonic spacing (Eq. 8.33) is every f ¼ 1:7 109 MHz.

The final step is to consider an ensemble of particles. Unlike thermal Bremsstrahlung

emission, the electron velocities do not follow a Maxwellian distribution (synchrotron

radiation is non-thermal). These relativistic particles comprise the electron component

of cosmic rays and so form a power law distribution in energy (see, e.g. Eq. 3.43) which

can be expressed in the form,

NðEÞ ¼ N0 EG ð8:34Þ

where NðEÞ (cgs units of cm3 erg1) is the number of cosmic ray electrons per unit

volume per unit interval of electron energy, N0 (cgs units of erg1 cm3) is a constant

(Ref. [116]) and G is the energy spectral index of the cosmic ray electrons (see Sect.

3.6.2). An integral over electron energy will return the total volume density of cosmic

ray electrons, NCRe,

NCRe ¼Z Emax

Emin

NðEÞ dE ð8:35Þ

where Emin, Emax are the minimum and maximum electron energies, respectively, that

are present in the relativistic gas.

Assuming that the overall distribution of electrons follows a power law as described

above and possesses an isotropic velocity distribution, the final emission coefficient,


j , and absorption coefficient, , can be determined. Provided G> 1=3, these are,

j ¼ c5ðGÞN0 B?Gþ1

2

2 c1

ðG 1Þ2

ð8:36Þ

¼ c6ðGÞN0 B?Gþ2

2

2 c1

ðGþ 4Þ2

ð8:37Þ

The constant, c1 and the functions, c5ðGÞ and c6ðGÞ are collectively (and loosely)

called Pacholczyk’s constants and are provided in Table 8.2. The absorption coefficient

is needed to account for the absorption of photons by the relativistic electrons within

the synchrotron-emitting cloud, a case called synchrotron self-absorption. The Source

Function Eq. (6.5)17 and optical depth (Eqs. 5.8, 5.9) are then, respectively,

S ¼ j

¼ c5ðGÞc6ðGÞ

B?1

2

2 c1

52

ð8:38Þ

¼ l ¼ c6ðGÞN0 B?Gþ2

2

2 c1

ðGþ 4Þ2

l ð8:39Þ

where l is the line-of-sight distance through the synchrotron-emitting source and we

have assumed no change in properties along the line of sight for Eq. (8.39). Since G is

positive for a typical cosmic ray energy distribution, Eq. (8.39) indicates that the

optical depth increases with decreasing frequency. Therefore a synchrotron-emitting

cloud becomes more opaque at lower frequencies.

Table 8.2 Pacholczyk’s constantsa

c1 ¼ 6:27 1018

G c5ðGÞ c6ðGÞ0.5 2:66 1022 1:62 1040

1.0 4:88 1023 1:18 1040

1.5 2:26 1023 9:69 1041

2.0 1:37 1023 8:61 1041

2.5 9:68 1024 8:10 1041

3.0 7:52 1024 7:97 1041

3.5 6:29 1024 8:16 1041

4.0 5:56 1024 8:55 1041

4.5 5:16 1024 9:24 1041

5.0 4:98 1024 1:03 1040

5.5 4:97 1024 1:16 1040

6.0 5:11 1024 1:34 1040

aRef. [116]. Note all quantities are in cgs units. G is the spectral index of the electron energy distribution givenby Eq. (8.34).

17Note that this is not an LTE situation so we cannot use Eq. (6.18) for the Source Function.


As we have done before, these equations can be put into the solution to the Equation

of Radiative Transfer to determine the synchrotron spectrum. Following the equations

in Sect. 6.3.5,

I ¼ S ð1 eÞ ðall Þ ð8:40Þ

I ¼ S / 52 ð 1Þ ð8:41Þ

I ¼ j l / ð 1Þ

2 / ð 1Þ ð8:42Þ

where

ðG 1Þ2

ð8:43Þ

is the frequency spectral index or just spectral index (not to be confused with the

absorption coefficient which is distinguished by the subscript, ). In Eqs.(8.41) and

(8.42), we have made a comparison to Eqs. (8.38) and (8.36), respectively, to see how

the spectrum varies with frequency. Synchrotron spectra for several values of Gdetermined from Eq. (8.40) are shown in Figure 8.14 along with the observed spectrum

Figure 8.14. Examples of the non-thermal synchrotron spectrum with ordinate (I) in cgs units.All curves with small black boxes are labelled with their electron energy spectral index, , and thecorresponding observed frequency spectral index, , given by Eq. (8.43). These curves have beencalculated from Eq. (8.40) using (in Eqs. 8.38 and 8.39): N0 ¼ 1010 ergr1 cm3, B ¼ 100mG,l ¼ 10 pc and values of the Pacholczyk’s constants from Table 8.2 that correspond to the value of used. The grey curve with larger boxes is the observed spectrum of the supernova remnant, Cas A(see image in Figure 1.2), using data from Ref. [11] and an angular diameter of 4:00


of the strongest radio source in the sky (aside from the Sun), the supernova remnant,

Cas A. Note that power laws are straight lines in this log–log plot.

8.5.3 Determining synchrotron source properties

Like any spectrum, that of synchrotron emission allows us to link the received radiation

to properties of the source. For simplicity, it is helpful to consider the optically thick

and optically thin parts of the spectrum separately.

As indicated by Eq. (8.41), if the emission is optically thick, the spectrum rises with

frequency (/ 5=2). This can be seen for the curve withG ¼ 3:0 in Figure 8.14 at the low

frequency end of the spectrum. In this part of the spectrum, the emission is described by

Eq. (8.38) which is a function only ofB? and some constants. This physical situation is not

so different from that of an optically thick thermal source for which only the temperature

can be determined. Since we cannot ‘see’ all the way through an optically thick synchro-

tron source, we cannot detect all the particles that are present in the relativistic gas.

Therefore, the brightness of the source is independent of particle density. An important

consequence is that a single measurement of emission from an optically thick synchrotron

source leads to a determination of the strength of the perpendicular magnetic field. For a

statistical sample of sources, on average we expect B? B to order of magnitude, so B?is sometimes simply referred to as the magnetic field strength. In reality, it is sometimes

difficult to find a source that shows a true optically thick synchrotron spectrum, however. It

is the compact sources, such as active galactic nuclei, that are most often optically thick

and these tend to consist of a number of CR populations with different power law energy

distributions and optical depths that, together, make a complex spectrum. For other

sources, the optically thick turnover may be at too low a frequency to be observed, or

else may be caused by free–free absorption from contaminating ionized gas (see Sect.

8.2.1), a situation that is likely to be the case for the supernova remnant, Cas A (Ref. [85],

see Example 8.6). If a true synchrotron self-absorption spectrum can be identified,

however, the magnetic field strength follows (Prob. 8.13).

As indicated by Eq. (8.42), if the emission is optically thin, the spectrum decreases as a

power law (/ ), several of which are shown in Figure 8.14. The power law slope in

frequency space, quantified by the spectral index, , is a characteristic of optically thin

synchrotron emission and is a consequence of the power law slope of the electron energy

distribution,G, the two being linked by Eq. (8.43). In the optically thin limit, the emission

is described by Eq. (8.42) with j given by Eq. (8.36). Thus, the emission depends on both

the particle density (via the constant, N0) and the magnetic field strength.

Example 8.6

The supernova remnant (SNR), Cas A (Figure 1.2), has an internal magnetic field of order afew mG (Ref. [8]). At what frequency will this SNR become optically thick to its ownsynchrotron radiation?

For the source to be optically thick implies that ¼ 1 so we need to solve Eq. (8.39) for

¼1. We let B? ¼ 2 103 G and, to obtain G, we measure the spectral index, , from


the optically thin part of Figure 8.14, finding ¼ 0:77. Using Eq. (8.43), this gives

G ¼ 2:54. From Table 8.2, Pacholczyk’s constants for this value of G are,

c1 ¼ 6:27 1018, c5 ¼ 9:51 1024, and c6 ¼ 8:09 1041 (assuming linear inter-

polation). For the line of sight distance, we assume that the depth through Cas A is

approximately equal to its diameter. From the caption to Figure 1.2, this is l ¼ 4 pc.

We now have all quantities required in Eq. (8.39) except N0.

To find N0, we must look at the optically thin part of the spectrum. From Figure 8.14, we

take the point at ¼ 1010 Hz and measure a value of I 4 1015 erg s1 cm2 Hz1

sr1. Using the applicable equation for the optically thin limit (i.e. Eq. 8.42 together with

Eq. 8.36 for j) gives, N0 ¼ 2:01 1013 erg1:54 cm3.

Finally, putting all required values into Eq. (8.39) and solving for frequency gives,

¼1 ¼ 8:3 MHz which is just to the left of the frequency range shown in Figure 8.1418.

Therefore, a more likely explanation for the observed higher frequency turnover is the

presence of absorbing ionized thermal gas.

As suggested above, for many sources, observations of pure synchrotron emission in

the optically thick limit are not always possible, whereas optically thin emission is

readily observed. This, then, poses a problem because emission of optically thin

radiation depends on both the magnetic field strength and the CR particle density.

We cannot obtain one without knowing the other. Moreover, the particle density, NCRe

Eq. (8.35), does not really give us the most important property of the relativistic gas.

There may be few CR electrons present, but the energy associated with those electrons

and other CR nuclei can be very high. If we wish to identify the original source that is

producing the relativistic gas, then some determination of the energy contained in the

relativistic gas and associated magnetic field is necessary.

The total energy represented by all CR particles and magnetic fields, is

UT ¼ UCR þ UB, where UCR includes both electrons as well as heavier nuclei and

UB is the energy contained in the magnetic field. Since it is the CR electron component

only that emits synchrotron radiation, the heavy particle component must be estimated.

It is common to represent the total CR energy, then, as UCR ¼ð1 þ kÞUe, where k is a

constant representing the ratio of energy contained in heavy nuclei compared to

electrons, and Ue is the energy of the CR electrons only. The constant, k is not always

well known and may vary in different environments, but is thought to be in the range,

40100.

The energy contained in the magnetic field increases with magnetic field strength

(see the equation for magnetic energy density in Table I.1). On the other hand, the

energy contained in the cosmic ray particles decreases with magnetic field strength.

This is a consequence of the fact that particles emit radiation more strongly (see Eq.

8.36), therefore losing energy faster, in higher magnetic fields. Thus, there is a

minimum in the function, UT when plotted against B. A conservative assumption is

18For a pure synchrotron spectrum, the frequency of the peak of the spectrum does not exactly coincide with thefrequency at which the spectrum becomes optically thick but, for Cas A, the corresponding peak would be about107 MHz.


therefore commonly made that the total energy in a synchrotron-emitting source is at or

near this minimum. The minimum energy values also turn out to be very close to those

of equipartition of energy, i.e. the condition in which the energy of the magnetic field is

equal to the energy in CR particles. By making the assumption of minimum energy, we

introduce one more constraint into the calculations which allows us to extract informa-

tion on both the magnetic field strength and CR particle energy from observations of

synchrotron radiation in the optically thin limit. This method is widely used to obtain

magnetic field strengths and energies or energy densities in synchrotron-emitting

objects. Even with the minimum energy assumption, the derived energies can be

extremely high, especially for distant extra-galactic radio sources (see next section).

Further details involving the minimum energy criterion and development can be found

in Ref. [116].

Like cyclotron emission, synchrotron emission is intrinsically polarized for a

magnetic field direction that is uniform. For synchrotron emission, it can be shown

that the maximum degree of polarization is Dp 70 per cent for 29G9 4 (Ref.

[177]). In fact, a detection of polarization is the best way of being sure that the emission

is indeed synchrotron. As indicated above, a power law slope is characteristic of

optically thin synchrotron emission, but the power law spectral index, , depends on

the energy injection spectral index, G (Eq. 8.43) which may vary depending on the

specific acceleration mechanism. In addition, some unresolved sources (like active

galactic nuclei) contain a mixture of different sources which result in the superimposi-

tion of various values of . The net result is that, although most optically thin spectra

are similar to those shown in Figure 8.14, it is possible for the observed value of to be

flat or even inverted (positive). If a flat slope were to exist, then this would mimic a

thermal Bremsstrahlung slope. The solution to the dilemma is a polarization observa-

tion. Since thermal Bremsstrahlung emission is never polarized, a clear polarization

detection decides the issue.

As is sometimes the case in nature, real situations may not be so clear cut. Most

synchrotron sources do show polarization. However, sources can contain both uniform

and random components to their magnetic fields. The interstellar medium in the Milky

Way, for example, has both uniform and random magnetic field components, each of

which is of order, a few m Gauss (Table 8.1). In other sources, the magnetic field lines

may become ‘tangled’ and there may be reduced (or no) directionality to the field

within the size scale of the telescope’s beam. This lowers the degree of polarization.

Thus, as indicated in Sect. 1.7, even strong radio jets tend to have Dp 9 15 per cent. A

clear detection of polarization is therefore indicative of synchrotron radiation but lack

of polarization does not rule it out. Very high brightness temperatures, however, are an

additional clue that the emission is synchrotron (see Sect. 8.1).

8.5.4 Synchrotron sources – spurs, bubbles, jets, lobes and relics

Most synchrotron-emitting sources are observed in the optically thin limit at

low frequencies where the emission is strongest (Figure 8.14). Although routine


measurements are now made for the stronger sources at optical and X-ray wavelengths,

synchrotron radiation is still most readily detected at radio frequencies. Since radio

waves do not suffer from dust extinction (see Appendix D.3), this also means that

synchrotron radiation is quite easily detected, passing through the ISM of the Milky

Way without significant attenuation. Unlike cosmic ray particles which are easily

scattered in the ISM (see Sect. 3.6.3), radio photons come directly from the location at

which they are generated, helping us to identify their origin.

Figure 8.15, for example, shows an all-sky image of ¼ 408 MHz emission which

contains predominatly synchrotron radiation. As discussed in Sect. 3.6.3, supernovae are

likely candidates for the acceleration of most Galactic cosmic rays at energies below the

‘knee’ of Figure 3.25 and their observed spectral indices of 0:899 0:2 imply

1:49G9 2:6 (Eq. 8.43) which spans the range, 2:19G0 9 2:5, believed to describe

the initial CR energy spectrum (Sect. 3.6.2). Two supernova remnants, Cas A and the

Crab Nebula, are marked in the figure, and the North Polar Spur is part of a local, 250 pc

diameter superbubblewhich may also have been formed by one or more supernovae (Ref.

[182]). This superbubble is likely related to the Local Hot Bubble discussed in Sect. 8.2.3.

Although supernova remnants are important sources of synchrotron emission, they

are by no means the only sources. In the realm of high energy astrophysics, there are

many objects with sufficient energies to accelerate electrons and sufficiently strong

magnetic fields to scatter them. In the Milky Way, these include shocks formed by

massive stellar winds, stellar jets, some binary stars, pulsars, and others. Synchrotron

emission is even observed from Jupiter and the Sun. Cosmic ray electrons that leak into

the ISM from discrete acceleration sites will also emit synchrotron radiation, since

408 MHz

Cas A

Cen ANorth Polar Spur

Cygnus A

LMC

CrabGC

Figure 8.15. All-sky map of synchrotron emission at ¼ 408 MHz in the same projection asshown in Figure 8.5. (The map is in false colour with brightest to dimmest represented by white/yellow through purple/deep blue and finally red.) Most emission from our own Milky Way is seenalong the Galactic plane as well as from spurs (e.g. the North Polar Spur) extending away from theplane. The Galactic Centre (GC) is marked, as well as the supernova remnants, Cas A and the CrabNebula. Most of the discrete sources away from the plane are extra-galactic, including the LargeMagellanic Cloud (LMC, a nearby galaxy), and the radio galaxies, Cygnus A and Centaurus A (for thelatter, the extended, double-lobed shape can be discerned) (Reproduced by permission of the Max-Planck-Institut fur Radioastronomie) (See colour plate)


magnetic fields are ubiquitous in the disk of the Milky Way. Typical ISM magnetic field

strengths of only a few mGauss are sufficient to generate observable emission. Thus,

much diffuse emission can be seen along the plane of the Galaxy in Figure 8.15.

Extra-galactic sources include active galactic nuclei (AGN) that are powered by

supermassive black holes (see Sect. 5.4.2 and Figure 5.5) and the bipolar jets that they

generate (e.g. Figure 1.4). The classic extra-galactic double-lobed radio sources were

the first discovered in the radio band and display two radio lobes at the ends of the jets,

as can be discerned for the radio galaxy, Centaurus A, in Figure 8.15. The powerful

radio galaxy, Cygnus A, is also identified in Figure 8.15 and is shown in more detail in

Figure 8.16. The inconspicuous elliptical galaxy at the centre, whose optical light

comes from stars, belies the true nature of its powerful core. The total energy contained

in its magnetic fields and cosmic ray particles, using the minimum energy assumption

(see previous section), is a spectacular 1060 erg (Ref. [123]). By comparison, the total

kinetic energy of a single supernova is 1051 erg. Not only are the particles that emit

synchrotron radiation relativistic, but the jets that feed the two lobes also have bulk

motions (meaning that the entire synchrotron-emitting plasma is moving in bulk) that

are relativistic, with velocities between 0:4 c and 1 c. The minimum energy magnetic

field strength in the radio lobes is, on average, 50 mG.

AGN activity is a strong contender for the origin of magnetic fields and cosmic rays

in the intergalactic medium of clusters of galaxies from which synchrotron emission

has also been detected. In some cases, regions of synchrotron emission are seen

DE

CL

INA

TIO

N (

J200

0)


19 59 32 30 28 26 24

40 44 30

15

00

43 45

30

Figure 8.16. The double-lobed extra-galactic radio source, Cygnus A (redshift, z ¼ 0:0562),shown in contours of ¼ 4:9 GHz emission, over the optical image (shown with black as highestintensity) of the same field of view. The faint greyish region at the centre of the image is thedistant elliptical galaxy that harbours a bright AGN at its centre. (Other black spots are stars in ourown Milky Way.) Two jets emerge from the AGN in opposite directions, ending in two large brightradio lobes on either side of the galaxy. The total flux density of the source is 213 Jy and theeffective FWHM of a single lobe is FWHM 10".


in the outskirts of galaxy clusters (for an optical image of a cluster of galaxies, see

Figure 8.8). These regions are called relics, or sometimes fossils or ghosts. The origin

of this emission is not entirely clear since these regions do not appear to be directly

connected to currently active galaxies within the cluster. Rather, the emission may be

generated within shocks that formed during the formation of the cluster and/or during

later cluster mergers. An older, existing population of cosmic ray particles that were

ejected from radio galaxies in the past could be reaccelerated in such shocks.

8.6 Inverse Compton radiation

Inverse Compton (IC) radiation is somewhat different from the other emission

discussed in this chapter because it results from the interaction of a particle with a

photon. However, since the emission begins with matter and ends with a photon

(i.e. matter ! photon ! photon) we include it here. As indicated in Sect. 5.1.2.2

and further detailed in Appendix D.2, Compton scattering occurs when a high energy

photon scatters inelastically from a free electron, boosting the kinetic energy of the

electron at the expense of the photon energy. IC radiation, often referred to as Inverse

Compton scattering, is the reverse of this process, as its name implies. It occurs when a

high energy electron inelastically scatters off of a low energy photon, the result being a

loss of kinetic energy for the electron and a gain of energy (a blueshift) of the photon.

The scattering will go in this direction if the electron energy is greater than the photon

energy in the centre-of-momentum rest frame of these two ‘particles’. For a highly

relativisitic electron and low energy photon, this rest frame is that of the moving

electron. For such a case, it can be shown that the frequency of the outgoing radiation,

as seen by a distant observer, is,

IC 2 ð h me c2Þ ð8:44Þ

where is the Lorentz factor Eq. (3.37) of the electron and is the frequency of the

photon before being scattered19. For example, for a relativistic electron with ¼ 104,

a radio photon, at a typical frequency of ¼ 1 GHz, would be ‘up-scattered’ to

IC ¼ 1017 Hz which is in the X-ray part of the spectrum (see Table G.6). The result

is a redistribution of photon energies with fewer photons in the radio and more photons

in the optical or X-ray (depending on ) than would otherwise be the case. In the

presence of a magnetic field, relativistic electrons will also emit synchrotron radiation,

so the addition of IC losses implies that the electrons will lose energy faster and the

radiated output over all will be higher than from synchrotron emission alone.

Any synchrotron-emitting source contains a plentiful supply of high-energy elec-

trons and emits low energy (e.g. radio) photons and these are also the required

ingredients for IC emission. Therefore, IC radiation, when observed, will be seen in

the same kinds of sources that emit synchrotron radiation. Because the relativistic

19Note that an electron moving with a Lorentz factor, , will ‘see’ a photon of energy, h in its own rest frame.

8.6 INVERSE COMPTON RADIATION 273

electrons have a power law distribution of energies, this IC radiation is non-thermal.

Moreover, the IC spectrum is also a power law and so is difficult to distinguish from

that of synchrotron radiation. However, not every synchrotron-emitting source is an IC

source as well. Synchrotron radiation depends on the strength of the magnetic field, B,

whereas IC radiation depends on the number of low energy photons that are available

for up-scattering. Thus, for a given number and distribution of relativistic electrons,

wherever B is strong and the radiation field is weak, synchrotron radiation will

dominate, and wherever the radiation field is strong and B is weak, IC radiation will

dominate. To be more precise, it can be shown that the fraction of the total radiative

luminosity of a source due to IC radiation, LIC, in comparison with the total radiative

luminosity due to synchrotron radiation, Ls, is (Ref. [155]),

LIC

Ls

¼ urad

uB

TBmax

1012

5 max

108:5

h ið8:45Þ

where urad is the energy density of the radiation field (see Sect. 1.4) and uB is the energy

density of the magnetic field (see Table I.1). The quantities, TBmax (K), and max (Hz),

are the brightness temperature and frequency, respectively, of the source at the peak

(the turnover) of the synchrotron spectrum, assuming that only synchrotron self-

absorption is producing the turn-over. Example 8.7 shows how the brightness tem-

perature and frequency dependence in this equation results.

Example 8.7

Show how the functional dependence on TB and in Eq. (8.45) is obtained.

From Eq. (1.16), the energy density of the radiation field, urad / I, where I is the

intensity of the source already integrated over frequency, so urad / I , where I is the

specific intensity. Using the definition of brightness temperature (Eq. 4.4) expressed in

the Rayleigh–Jeans limit Eq. (4.6) since most of the photons will be in the radio part of the

spectrum and h k TB, we have, I / TB 2, which leads to, urad / TB 3.

Now, from Table I.1, uB / B2. In the optically thick limit, the magnetic field strength is

related to the specific intensity via Eqs. (8.41) and (8.38) and this dependence should still

be approximately correct at the peak of the curve where the spectrum turns over. Letting

B B?, and rearranging Eq. (8.38) to solve for B gives B / 5=I2. Thus, uB / 10=I

4.

Now, converting the specific intensity to a brightness temperature as we did above, then,

uB / 2=TB4.

Finally, from the above two results, the ratio, urad=uB / ðTB 3Þ=ð2=TB4Þ / TB

5 which is consistent with Eq. (8.45).

The importance of Eq. (8.45) is in the extremely strong dependence of IC losses on

the brightness temperature of the source. IC radiation should not be very important

until the brightness temperature approaches 1012 K at the reference frequency given in

the equation (e.g. Prob. 8.14). Once brightness temperatures exceed 1012 K, however, a


situation called the Compton catastrophe, the electrons should lose energy so rapidly

from IC losses (of order days, Ref. [132]) that no sources with such high brightness

temperatures should actually exist. Thus Eq. (8.45) suggests a natural upper limit of

TB 1012 K for source brightness temperatures20. In fact, this limit appears to be

approximately correct when compared to observations. There are sources in which

higher brightness temperatures are observed, but these are moving sources which were

not considered when deriving Eq. (8.45). If the entire synchrotron-emitting plasma has

a bulk motion that is moving relativistically towards the observer, such as in a jet

pointed towards the observer, then the emission will be brighter than if the plasma were

stationary, a situation called Doppler beaming.

The most famous example of inverse Compton scattering is the Sunyaev–Zeldovich

effect (S–Z effect). This is the IC up-scattering of 2:7 K CMB photons by very hot (but

non-relativistic) electrons in clusters of galaxies, shifting the CMB photons to higher

frequencies (see Prob. 8.15). An abundant supply of ionized gas, and therefore hot

electrons, is present in some galaxy clusters, as illustrated in Figure 8.8, and back-

ground photons will pass through such clusters en route to us. The result is a distortion

of the 2:7 K CMB spectrum. Rather than the ‘perfect’ black body that we saw in

Figure 4.3, a distorted spectrum, such as shown in Figure 8.17 results. Although the

effect is typically quite small (T < 1 mK), it has now been measured in many

galaxy clusters. The importance of the effect is that the observed shift depends on the

electron density and temperature (ne;Te). Since X-ray measurements of thermal

Figure 8.17. Illustration of the Sunyaev–Zeldovich effect, showing the undistorted CMB spectrum(dashed curve) and the spectrum after CMB photons have passed through the ionized gas in acluster of galaxies (solid curve). The cluster has been made 1000 more massive than a typicalcluster to exaggerate the effect. (Reproduced by permission of John Carlstrom, Carlstrom, J. E.,Holder, G. P., and Reese, E. D., 2002, ARAA, 40, 643)

20Other considerations related to equipartition of energy may make the upper limit somewhat smaller, of orderTB 1011 K (see Ref. [132]).


Bremsstrahlung emission also depend on ne and Te, the S–Z effect provides an

independent check on these values. More importantly, if X-ray data are available

and Te is known (Sect. 8.2.3), combining all information allows the determination of

the distance to the cluster. Knowledge of the distance and total gas mass (via Eq. 8.9)

leads to crucial data on the cosmological parameters (see Sect. 8.2.3, Figure 3.4).

Problems

8.1 (a) Determine the root-mean-square speed, vrms, and corresponding kinetic energy,

Ek, of an electron in an HII region of electron temperature, Te ¼ 104 K.

(b) If the electron lost all of its kinetic energy (came to a stop) during an encounter with

a nucleus without recombining, what would be the frequency of the resulting emission and

in what part of the spectrum does this frequency occur?

(c) Compare the frequency of part (b) with a typical thermal Bremsstrahlung photon at a

radio frequency of ¼ 5 109 Hz from an HII region. Based on this comparison, what

fraction of an electron’s kinetic energy has actually been lost during an encounter with a

nucleus?

8.2 The spectrum of an HII region is observed to have an exponential cutoff such that the

emission declines by a factor, 1=e (its e-folding value), at a wavelength of 2:4 mm. What is

the temperature of this HII region?

8.3 With the help of a spreadsheet or computer algebra software, determine and plot the

spectrum of free–free emission from the Lagoon Nebula shown in Figure 3.13. Ensure that

the low frequency optically thick part of the spectrum and the high frequency cutoff are

shown. For simplicity, assume that the ionized gas consists of pure hydrogen and that the

Gaunt factor is constant with frequency.

8.4 (a) Show that h k Te for any HII region observed at radio wavelengths.

(b) Show that Eq. (8.12) results from Eq. (8.4).

8.5 For the HII region, IC 5146, whose radio emission, as shown in Figure 8.4, is in the

optically thin limit, find the following:

(a) The frequency at which the HII region becomes optically thick.

(b) The electron density, ne, ionized hydrogen mass, MHII, total gas mass, Mg, and

excitation parameter, U.

(c) The radius of its Stromgren sphere (assume that the effects of dust are negligible).

(d) The number of ionizing photons per second, Ni, from the star, BDþ463474.

8.6 Using the parameters of the hot intracluster gas given in Figure 8.3, determine the

frequency at which the emission becomes optically thick. In what part of the spectrum does

this occur and is it observable from the ground?


8.7 Show that the luminosity within a frequency band, 1 to 2, is given by Eq. (8.19) for

optically thin thermal Bremsstrahlung radiation from ionized hydrogen gas of constant

temperature and density.

8.8 For the cluster of galaxies, Abell 2256, whose soft X-ray emission is shown in

Figure 8.8, determine or estimate the following:

(a) Its distance, D (Mpc).

(b) The volume, V (kpc3), of the X-ray emitting gas, assuming that it occupies the entire

volume of the cluster.

(c) The gas temperature, Te (K).

(d) The mean electron density, ne (cm3).

(e) The ionized hydrogen mass, MHII (M).

(f) The fraction, Mgas=Mtot, where Mgas is the total gas mass and Mtot is the total

light+dark mass of the cluster.

8.9 (a) Write an expression for the conservation of energy for an electron which

makes a transition from an initial free state in an ionized hydrogen gas to a final bound

state, showing that the difference in energy corresponds to the energy of the emitted

photon.

(b) Compute the frequency of an emitted photon for a transition to the n ¼ 1 and n ¼ 2

levels of hydrogen when (i) the initial electron velocity is zero, and (ii) the initial electron

velocity is the most probable velocity in a gas at Te ¼ 1:58 105 K.

(c) Compare the results of part (b) to the curve of gfb shown in Figure 8.2 and explain the

shape of this curve.

8.10 Cyclotron emission from electrons moving at v 2 104 km s1 in a region of

the magnetosphere of Jupiter is observed at the gyrofrequency, 0 ¼ 540 kHz.

(a) Determine the magnetic field in this region.

(b) Find the gyroradius of the electrons. Is this larger or smaller than the wavelength of

the emitted radiation?

(c) Write an expression for 0=p in terms of B and ne with constants evaluated. What is

the maximum possible electron density of this region?

(d) What is the gyrofrequency of a proton, op, in the same region? Repeat the first part

of step (c) above using op and determine the maximum possible electron density that

would allow cyclotron emission from a proton to escape from the region.

8.11 Suppose a Jupiter-like planet is orbiting a Sun-like star and that this stellar system is

at a distance of 10 pc from the Earth. The system is observed at ¼ 30 MHz and is

unresolved. Adopt TB (30 MHz) ¼ 5 105 K (Sect. 4.1.1).


(a) What is the flux density of the star (cgs units) as measured at the Earth? Assume that

the star is not undergoing any radio bursts.

(b) From the information given in Figure 8.12, what is the flux density of the planet (cgs)

as measured at the Earth? Assume that the planet is undergoing a radio burst, and has a

modulating moon like Io.

(c) Comment on the prospects of measuring radio bursts from extrasolar planets at low

frequencies and the possibility of detecting extrasolar planets via their radio emission

assuming that the telescope is sensitive enough to detect the stellar radio emission.

8.12 Determine the relativistic gyroradius (kpc) of a CR proton that has the highest

energy shown in Figure 3.25 in the ISM of the Milky Way. Compare this to the distance of

the Large Magellanic Cloud (see Prob. 7.16 for data) and comment on the ability of

galaxies, in general, to retain such high energy cosmic rays.

8.13 In 1993, a supernova (SN1993J) was observed in the nearby galaxy, M 81, which is a

distance D ¼ 3:6 Mpc from us. At a time, 273 days after the observed explosion21, the

following parameters (approximated from Ref. [122]) described the radio emission from

this source: (i) a rising spectrum (I / 5=2) at low frequencies with a flux density of

f ¼ 72 mJy at a frequency of ¼ 1:43 GHz, (ii) a falling spectrum (I / 1) with a

flux density of f ¼ 25 mJy at a frequency of ¼ 23 GHz, (iii) a radius of r ¼ 0:0123 pc,

and (iv) a minimum energy cutoff to the electron energy spectrum of Emin ¼ 46 MeV.

Determine the following parameters for this supernova (all in cgs units):

(a) The electron energy spectral index, G.

(b) The solid angle subtended by the source, .

(c) The brightness temperature at ¼ 1:43 GHz, TB.

(d) The strength of the perpendicular magnetic field, B?.

(e) The constant of the electron energy spectrum, N0.

(f) The number density of cosmic ray electrons, NCRe.

8.14 The total flux density of Cygnus A (see Figure 8.16) is f ¼ 2:19 104 Jy at

¼ 12:6 MHz, divided roughly equally between the two radio lobes. Assuming that this

flux density represents the peak of the spectrum of Cygnus A and that synchrotron self-

absorption is the only absorption process that is producing the spectral turnover, determine

the ratio of LIC=Ls for one radio lobe. How important is inverse Compton radiation in

comparison to synchrotron radiation for this source?

8.15 Verify that the condition in parentheses in Eq. (8.44) is satisfied for the case of CMB

photons and hot intracluster gas such as is shown in Figure 8.8.

21The actual explosion occurred 11.7 million years prior to its observation in 1993 since M 81 is 3.6 Mpc ¼ 11:7million light years away.


9Line Emission

Have you seen anything so beautiful?

– C. V. Raman, pointing to the evening sky (Ref. [128])

By the end of the 19th century, the German physicist, Max Planck, had explained the

continuous black body spectrum by allowing light to exist as a photon, or ‘particle’ (see

the Appendix at the end of Chapter 4). His theoretical development introduced the

constant, h, now called Planck’s constant. The concept of quantized light was further

strengthened when Einstein, in 1905, successfully explained the photoelectric effect –

the release of electrons in some metals and semiconductors via the action of incident

light – by treating light as a particle. Still not understood, however, was the series of

lines seen in the spectrum of the hydrogen atom. It was the Danish physicist, Niels

Bohr, who brought the concept of quantization to the realm of the atom. He proposed,

in 1913, that electrons did not radiate unless they made a transition from one state to

another. The difference in energy between these states could be related to the emitted

energy of a photon, according to E ¼ h n. This profound shift from the classical view

of requiring orbiting electrons to radiate, to one in which radiation results only from

transitions between states, won Bohr the Nobel Prize in 1922. It also set the scene for

Heisenberg, Sommerfeld, Schrodinger, Pauli and others to further develop and refine

the new field of quantum mechanics.

This chapter is devoted to an understanding of the observed spectral lines from

atoms and molecules that result from discrete internal changes in energy. While the

details of quantum theory are beyond the scope of this book, an introduction to

quantum mechanical principles for the hydrogen atom has been provided in

Appendix C. Much of the groundwork has been laid out earlier. For example, we

have seen that the population of bound states in an atom under LTE conditions is

described by the Boltzmann Equation (Sect. 5.1.1.2, 5.1.1.3, 5.1.2, and Appendixes

D.1.2, D.1.3, D.1.4, and D.1.5. Bound–bound absorption was mentioned in Sect. 5.2.1


and recombination lines were introduced in Sect. 3.4.7. The conditions for forming

absorption and emission lines were described in Sect. 6.4.2, and the information that

can be obtained from detecting the l 21 cm line of hydrogen in both emission and

absorption was presented in Sect. 6.4.3. In Sect. 7.2, we also discussed how the

relatively simple observation of a redshift can lead to some very powerful conclusions,

from information about black hole masses to an understanding of the nature of our

expanding Universe.

We focus now on what spectral lines can be formed in atoms and molecules and how

these spectral lines provide us with information about the sources that emit them.

Although a line spectrum can form from pure cyclotron emission (Sect. 8.5.1), we will

restrict the discussion to spectral lines that are formed from discrete quantum transi-

tions in atoms and molecules as well as their ions and isotopes, that is, bound–bound

transitions. We also focus on the line emission process and therefore will deal with

‘downwards’ (higher to lower energy) transitions, ignoring the possible presence of a

background source1. The latter can be handled by considering the appropriate solution

to the Equation of Transfer, as described in Chapter 6.

9.1 The richness of the spectrum – radio waves to gamma rays

What kinds of bound–bound transitions are possible? These depend on the quantiza-

tions that occur within atoms and molecules The possibilities are: electronic transi-

tions, vibrational transitions, rotational transitions, and nuclear transitions, all of

which are quantized.

9.1.1 Electronic transitions – optical and UV lines

Electronic transitions occur when an electron changes from a higher to lower energy

state in atoms and molecules or their ions and isotopes. These are the familiar

transitions that have already been discussed in various contexts. For example, a rich

variety of spectral lines from many different species was seen in the optical spectrum of

an H II region in Figure 8.9.

The electronic transitions of hydrogen are discussed in Appendix C. If the hydrogen

is ionized, a recombination line spectrum is observed (Sect. 3.4.7) as electrons

continuously recombine with protons and then cascade down various bound–bound

levels. The highest frequency line that is possible from the hydrogen atom results from

the largest energy difference between bound–bound states. This would be close to the

ionization energy (see Eq. C.7) of 13:6 eV for which the corresponding wavelength,

l 91:18 nm, is in the ultraviolet part of the spectrum. At the other extreme, transitions

between very high quantum levels result in radio wave emission since the energy levels

1The 2:7 K CMB background will always be present, but we will assume that its brightness is negligible, incomparison to the line.

280 CH9 LINE EMISSION

of hydrogen become progressively closer together with increasing n (see Figure C.1).

Thus, line emission from hydrogen progressively shifts in frequency across the

electromagnetic spectrum from the UV to the radio band, depending on which

principal quantum numbers are involved. At higher quantum numbers, however, the

transition probabilities are lower (e.g. see values of Aj;i in Table C.1), so it is common to

refer to electronic transitions in atoms to be ‘typically’ in the optical or UV parts of the

spectrum.

Electronic transitions occur in other atoms and in molecules as well. When all

species are taken into account, spectral lines are seen across the entire electromagnetic

spectrum from the X-ray to radio bands. X-ray lines have been observed from the

highly ionized iron ions in a Solar flare, for example (Figure 6.7). Larger, heavier atoms

can have higher energy transitions and also more of them, since there are more

electrons associated with such elements. Put together, a wide array of possible

electronic transitions are available over a broad waveband, though only a subset of

these may be excited at any time, depending on the physical conditions in the gas.

Spectral lines are therefore probes of the physical conditions in the gas, as we will soon

see. At the very least, the identification of spectral lines as belonging to certain species

provides information as to which elements are present in the gas.

9.1.2 Rotational and vibrational transitions –molecules, IR andmm-wavespectra

In molecules, there are two additional types of transitions that can occur, besides

electronic ones. These are rotational transitions and vibrational transitions. Rotation

and vibration are illustrated for a diatomic molecule in Figure 9.1, but it should be

noted that there are no exact quantum mechanical analogues to these motions. Just as

we saw for an electron in ‘orbit’ about the nucleus (Appendix C.1) and for the ‘spin’ of

an electron (Appendix C.2), the rotation and vibration of a molecule are convenient

models that allow us to picture these new energy levels and to derive their energies

from a classical starting point.

If a molecule rotates, it has some energy associated with that rotation. A good

classical approximation is that of a rigid rotator which, for a diatomic molecule, would

be as if the molecule were a dumbbell with a nucleus at either end. As illustrated in

Figure 9.1.a, the rotation is about either of two principal axes2. If the two nuclei have

different atomic weights, then the molecule will have a permanent electric dipole

moment (Table I.1) due to the way in which charge is distributed in the molecule. For

example, in the CO molecule, the oxygen atom has a slight negative charge and the

carbon atom has a slight positive charge. In the classical picture, as the molecule

rotates, so does the dipole moment and this would produce electromagnetic radiation

2Rotation about the third axis is not energetically important since the moment of intertia, I, is very small. For anyobject consisting of individual particles, I ¼

Pmi ri

2, where mi is the mass of particle, i, and ri is its distancefrom the axis of rotation.

9.1 THE RICHNESS OF THE SPECTRUM – RADIO WAVES TO GAMMA RAYS 281

continuously. However, just as we saw for ‘orbiting’ electrons, quantum mechanically,

this is not the case. The rotation is quantized and radiation is emitted only when a

transition occurs from one rotational state to another. It is nevertheless necessary for an

electric dipole to be present in order for a rotational change to produce a rotational

spectral line. Molecules like H2, N2, and O2, for example, do not have electric dipole

moments since their charge is equally distributed, and therefore these molecules also

display no rotationally-induced electric dipole radiation. These molecules may have

electric quadrupoles, however3, and if so, much weaker (by many orders of magnitude)

rotational lines can result from changes in the quadrupole moment. As can be seen in

the potential energy diagram of Figure 9.1.c, rotational transitions correspond to the

smallest energy changes in molecules and therefore these lines have the lowest

frequencies, typically in the far-IR, sub-mm or mm parts of the spectrum.

Aside from rotation, a molecule can also vibrate, as illustrated in Figure 9.1.b and a

classical analogue is that of simple harmonic motion. As with rotation, the vibrational

states are quantized and a spectral line is emitted only if there are changes between

these states. The close-to-parabolic shape of simple harmonic motion can be seen in the

potential energy diagram of Figure 9.1.c. If the positive nuclei are separated by a small

distance, there is little room for electrons between them and the force between the two

protons is repulsive. At a large distance, more electrons will occupy the internuclear

Centerof mass

Centerof mass

Centerof mass

(a) (c)

interatomic distance

(b)

x x

+

+ –

+

–

–

z

y y

z V

Figure 9.1. Molecular transitions in a diatomic molecule. (a) Illustration of rotation. Rotation isin the x–y plane at left, and in the y–z plane at right. The rotation is quantized and a spectral line isemitted only when there is a transition between two different rotational states. (b) Illustration ofvibration. The motion resembles that of a simple harmonic oscillator. (c) A sample energy leveldiagram showing the molecule’s potential energy (including all nuclei and electrons) as a functionof separation between nuclei. Electronic, vibrational and rotational transitions are shown.

3Taking a simplified case in which the electrons and protons are along a straight line, the electric dipole moment isp1 ¼

Pqi xi, where qi is the charge of particle, i, and xi is its position. The electric quadrupole moment is then

p2 ¼P

qi xi2. Even though the electric dipole moment may be zero, the electric quadrupole moment may be non-

zero, depending on the configuration.


space and the force will be attractive. Thus, the molecule can oscillate over different

internuclear separations if perturbed. If the separation becomes too great (to the

right on the figure) the molecule will dissociate, but if the separation is close to the

minimum of the curve for any given electronic state, then it is stable4. Transitions

between vibrational states are of much higher energy (typically in the IR) than those

between rotational states, the molecule vibrating perhaps 1000 times during a single

rotation.

In reality, a single transition may involve a change in both vibrational and rotational

states and these are called ro-vibrational transitions. The changes in quantum numbers

must obey the same quantum mechanical selection rules as if they occurred individu-

ally. The total change in energy is the sum of the changes in the vibrational and

rotational energies, Ero-vib ¼ Evib þErot, but this will be dominated by Evib

with Erot small in comparison. Observationally, for a given vibrational transition, all

possible allowed rotational transitions also occur and therefore the vibrational line

is broken into discrete components with spacing of the rotational steps, i.e. the

rotational spectrum is superimposed on the vibrational line. Since the rotational steps

are very closely spaced, this spreads out the vibrational ‘line’ into a spectral band.

Similarly, it is possible for a transition to occur that involves a change in all three

electronic, vibrational, and rotational quantum numbers in one transition. In such a

case, the total energy change is the sum of all three, with the electronic energy change

dominant.

From the above discussion, it is clear that the observed spectrum of a single molecule

will, in general, be far more complex than the spectrum of a single atom. A molecular

cloud, moreover, will contain a variety of different kinds of neutral molecules as well as

charged molecules and molecules containing isotopes, each of which has its own

spectrum. Put together, molecular spectra tend to be so rich that spectral surveys are

undertaken to search for and identify the lines in molecular clouds. Figure 9.2 shows a

dramatic example. The Orion KL5 nebula contains a massive, young stellar cluster as

well as abundant dust and molecular gas. The molecular gas reaches densities of

n 107 cm3 and temperatures up to T 200 K in the core of the cluster (Ref. [98]).

These conditions differ from those of typical interstellar molecular clouds that are far

from star-forming regions (n 103 cm3, T 20 K, see Table 3.1). Thus, the excita-

tion conditions are such that more lines are seen in the spectrum of Orion KL than

would be the case in colder ISM clouds. Figure 9.2 shows only a tiny fraction of the

lines that are actually present in this nebula. Another survey of Orion KL over the

frequency range, 790 GHz (0:38 mm) to 900 GHz (0:033 mm), for example, reveals

approximately 1000 spectral lines of which about 90 per cent have been identified

(Ref. [43]).

How many different molecules have actually been detected in astronomical objects?

Table 9.1, which provides a list to date, indicates that more than 130 molecules have so

4The minimum energy state is not exactly at the minimum of the curve. Even the ground state will have a slightvibration associated with it.5‘KL’ stands for ‘Kleinmann–Low’.


Figu

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Table 9.1. Molecules detected in astronomical sourcesa

2 3 4 5 6 7

H2 C3 c-C3H C5

C5H C6H

AlF C2H l-C3H C4H l-H2C4 CH2CHCN

AlCl C2O C3N C4Si C2H4 CH3C2H

C2 C2S C3O l-C3H2 CH3CN HC5N

CH CH2 C3S c-C3H2 CH3NC CH3CHO

CHþ HCN C2H2 H2CCN CH3OH CH3NH2

CN HCO NH3 CH4 CH3SH c-C2H4O

CO HCOþ HCCN HC3N HC3NHþ H2CCHOH

COþ HCSþ HCNHþ HC2NC HC2CHO

CP HOCþ HNCO HCOOH NH2CHO

SiC H2O HNCS H2CNH C5N

HCl H2S HOCOþ H2C2O l-HC4H(?)

KCl HNC H2CO H2NCN l-HC4N

NH HNO H2CN HNC3 c-H2C3O

NO MgCN H2CS SiH4 H2CCNH(?)

NS MgNC H3Oþ H2COHþ

NaCl N2Hþ c-SiC3

OH N2O CH3

PN NaCN

SO OCS

SOþ SO2

SiN c-SiC2

SiO CO2

SiS NH2

CS H3þ

HF H2Dþ, HD2þ

SH SiCN

HD AlNC

FeO(?) SiNC

O2(?)

CFþ

SiH(?)

8 9 10 11 12 13

CH3C3N CH3C4H CH3C5N HC9N C6H6(?) HC11N

HCOOCH3 CH3CH2CN (CH3)2CO CH3C6H C2H5OCH3(?)

CH3COOH (CH3)2O (CH2OH)2

C7H CH3CH2OH CH3CH2CHO

H2C6 HC7N

CH2OHCHO C8H

l-HC6H(?) CH3C(O)NH2

CH2CHCHO(?)

CH2CCHCN

aSee the table of S. Thorwirth from data provided by S. Martin at http://www.ph1.uni-koeln.de/vorhersagen/molecules/main_molecules.html. Numbers at the tops of the columns specify the number of atoms. A * specifiesmolecules that have been detected by their ro-vibrational spectrum. A ** specifies molecules that have beendetected by electronic spectroscopy only. A (?) indicates a probable detection.


far been identified. The largest in this table is HC11N, containing 13 atoms. However,

as shown in Figure 3.24, if the nature of polycyclic aromatic hydrocarbons (PAHs,

Sect. 3.5.2) is confirmed, the true number is higher and so is the number of atoms

contained within the molecules.

9.1.3 Nuclear transitions – c-rays and high energy events

Probing still deeper into the heart of an atom’s structure brings us to the nucleus within

which further quantization occurs. If a nucleus finds itself in an excited state as a result

of a radioactive decay or high energy collision, then it may de-excite with the emission

of a g-ray photon at a specific energy. Various decay chains and daughter products are

possible, depending on the specific nucleus and its stability. A mathematical descrip-

tion of nuclei and their energy levels is more difficult than has been discussed for the

electronic, vibrational and rotational transitions. Nevertheless, many nuclear transi-

tions are well known. Table 9.2 lists some that are important in astrophysics, indicating

the relevant decay chains and energies of the emitted g-rays. Those that have been

detected are noted in bold-face type. Different isotopes of the same species produce

lines at different energies, so it is possible to distinguish, not only between the element

that is producing the line, but also which isotope is involved.

Gamma-rays have the highest energies known for light (Table G.6) so the original

energy required to excite the nucleus must also have been of the same order. Thus, g-ray

emission is associated with high energy events. Examples are solar flares, novae and

supernovae (Sect. 3.3.3), and spallation of nuclei (Sects. 3.3.4, 3.6.1) by the high

energy cosmic rays that permeate the Galaxy. Massive stellar winds from Wolf–Rayet

stars (Sect. 3.6.1 and Figure 5.10) can also deposit nuclei that have been created via

nucleosynthesis in their stellar interiors into the ISM. Energies from nuclear transitions

Table 9.2. Some important nuclear g-ray Linesa

Isotope Mean lifetimeb Decay chainc Energyd(keV)

7Be 77 days 7Be! 7Li 47856Ni 111 dayse 56Ni! 56Co! 56Feþ eþ 847, 1238

2598, 177157Co 1:1 years 57Co! 57Fe 122, 13622Na 3:8 years 22Na! 22Neþ eþ 127544Ti 85 years 44Ti! 44Sc! 44Caþ eþ 1157, 78, 6826Al 1:04 106 years 26Al! 26Mgþ eþ 180960Fe 2:2 106 years 60Fe! 60Co 1173, 1332

aRef. [48].bThis ‘lifetime’, t, is related to the half-life, t1=2, by t1=2 ¼ t ln 2. In the event of two decays, the longer time isgiven.cNot all possible decay routes are shown – just those that are relevant in the production of the g-rays indicated.Decays that result in the emission of a positron (eþ) are also indicated.dValues in bold-face have been detected.eThe 56Ni! 56Co part of this chain has a lifetime of only 8:8 days.


are typically in the MeV range which, although considered high energy, are at the lower

end of the full g-ray spectrum observed in nature.

Supernovae are a particularly important source of g-ray lines since the nucleosynth-

esis that occurs during a supernova explosion leaves many nuclei in excited and

unstable states. The g-rays that result from de-exciting nuclei have been shown to

drive supernova light curves which, like radioactivity, decay exponentially with time

(Sect. 3.3.3). For example, the bolometric (Sect. 1.1) light curve of Supernova 1987A6,

which occurred in the nearby galaxy, the Large Magellanic Cloud, decayed with a half-

life of 77 days, nicely corresponding to the known 77 day half-life (111 day lifetime7)

of the decay of 56Co to 56Fe. The g-ray photons that are emitted during this decay

interact with the surrounding gaseous material via Compton scattering (Sect. 5.1.2.2)

and, with further interactions, the energy is converted to the observed bolometric

luminosity8. SN1987A has now been observed for 20 years and the rate of decline of its

light curve has changed as the quantity of 56Co has declined and other isotopes with

longer half-lives have become more important sources of g-ray photons. The current

light curve is well-matched to the decay rate of 44Ti, although a direct detection of the

associated g-ray lines has yet to be made (Ref. [48]).

Looking at the Milky Way, we can also see g-ray line emission from Galactic

supernovae. In the young ( 312 yr old) nearby (d ¼ 3:4 kpc) supernova remnant

(SNR), Cas A (Figure 1.2), the 1157 keV g-ray line from 44Ti has indeed been detected

(Ref. [78]), even though this isotope has a half-life of only 59 yr (lifetime of 85 yr,

Table 9.2). From the line flux, it has been possible to measure the total mass of 44Ti

( 2 104 M) produced in the supernova explosion. Such studies not only con-

tribute to our knowledge of elemental abundances (see Figure 3.9) but also provide a

better understanding of the processes and conditions that occur during the SN explo-

sion itself. Spontaneous nuclear de-excitations, themselves, are not sensitive to the

temperature and density of the surrounding material (they depend on nuclear proper-

ties), but the rate at which nucleosynthesis can form these nuclei will be sensitive to

physical conditions.

The 44Ti g-ray lifetime is not that different from the expected mean time between

supernova explosions in our Milky Way (about one every 50 yr, Sect. 3.3.4). If this line

is detected and originates in supernovae, then the sources from which the line is

observed should appear discrete. On the other hand, if a g-ray line from a longer lived

isotope of supernova origin is observed, then its distribution should describe the

distribution of Galactic supernovae integrated over a timescale corresponding to the

lifetime of the line. This is the case for the 1:809 MeV line from 26Al which has a

lifetime of a million years (Table 9.2). Over this period of time, with a mean supernova

rate of one per 50 yr, we might expect to see the integrated g-ray luminosity from

approximately 20 000 supernovae (SNe). These SNe should originate from different

6This supernova, discovered in 1987 from Chile by Ian Shelton of the University of Toronto, Canada, has played apivotal role in testing supernova models.7The ‘lifetime’, t, is related to the half-life, t1=2, by t1=2 ¼ t ln 2.8This process is reminiscent of the diffusion and energy degradation of photons in the Sun as they travel from thecore to the surface (see Sect. 3.4.4).


locations in the Galaxy, leading to distributed, rather than discrete, emission. More-

over, since the lifetimes of the massive progenitor stars are also of order, 106 yr, the

observed 26Al emission will only be seen in locations where massive star formation has

recently occurred. This line is therefore a tracer of massive star formation. We have

encountered such tracers before (e.g. H II regions, Sect. 3.3.4), but compared to optical

observations, g-rays have an advantage in that they are highly penetrating and can

travel through the ISM virtually unhindered.

A plot of the 26Al distribution in the Milky Way is shown in Figure 9.3. Not all the

emission in this figure results from supernovae alone. Between 20 and 50 per cent may

originate from Wolf–Rayet stars (Ref. [117]), but since Wolf–Rayet stars, themselves,

are massive and short-lived, the bulk of the 26Al emission still traces sites of massive

star formation in the Galaxy. Doppler shifts of this line have also provided information

on motions of the various sources (Sect. 7.2.1). Even the broad-scale rotation of the

Galaxy has been measured from this line emission (Ref. [49]).

9.2 The line strengths, thermalization, and the critical gasdensity

In Chapter 8, we provided the emission coefficient, jn, for continuum radiation and then

found the specific intensity, In, by applying the relevant solution to the Equation of

Figure 9.3. Map of the 26Al 1:809 MeV g-ray line emission in the Galaxy shown in a Hammer–Aitoff projection (e.g. see Figure 3.2 or Figure 8.5) with the Galactic Centre at the map centre. Thismap (see Ref. [125]) represents 9 yr of data collected by NASA’s Compton Gamma Ray Observatoryusing the imaging Compton Telescope (COMPTEL). (Reproduced by permission of R. Diehl, Plueschkeet al., 2001, in Exploring the Gamma-Ray Universe (4th Integral Workshop), Eds. A. Gimenez,V. Reglero and C. Winkler, ESA SP-459, Noordwijk, 56–58)


Transfer. Other quantities, such as flux density or luminosity, followed. The approach

for spectral lines is essentially the same, but with the additional constraint that the line

is present over a well-defined frequency interval. In order to characterize some line

strength, then, we could either refer to the peak value of the line, or we could refer to

the integral over the line. Both of these quantities are physically meaningful but, since

the integral includes the total number of particles that are contributing to the line,

we will use the integral to specify the ‘line strength’. The line strength is determined by

the downwards transition rate of whatever mechanism is producing the line, as well as

the number of particles that are in the upper level.

Downwards transitions can occur spontaneously with a rate given by the Einstein A

coefficient for a given transition (e.g. Table C.2), or can be induced via a collision,

whose rate depends on the collision cross-section for that particlar transition as well as

various physical parameters such as the density and temperature of the gas (see Eqs.

3.19 and 3.7)9. Example 5.1 provided a comparison of these two rates for the Ly a line

of hydrogen under one set of conditions, showing the tendency of this line to undergo a

spontaneous de-excitation, given the atom’s short natural lifetime in the n ¼ 2 state. On

the other hand, the l 21 cm line of H I is collisionally induced, since it has a longer

spontaneous de-excitation timescale in comparison to a typical collision time. Since

there is a different Einstein A coefficient and a different collision cross-section for each

line, the probability of a downwards transition must be considered, spectral line by

spectral line, for any given species under a set of physical conditions.

As discussed in Sects. 3.4.4 and 3.4.5, if all transitions of an atom are collisionally-

induced, then the gas is in LTE. We now allow for the possibility that some transitions

may be collisionally-induced and some may not – a non-LTE situation. However, if a

specific transition is collisionally-induced, then it can be stated that the line is in LTE

(see also arguments in Sect. 8.2.1). In such a case, the Boltzmann Equation (Eq. 3.23)

holds for the individual line being considered. The line is then said to be thermalized

and the LTE solution of the Equation of Transfer (Eq. 6.19) applies for the line.

There is some importance in knowing whether a line results from a collisional or a

spontaneous downwards transition because, as outlined in Sect. 3.4.4, only if collisions

are somehow involved can the observed line radiation tell us something about the gas

kinetic temperature. The gas density at which the downwards collisional transition rate

equals the downwards spontaneous rate for a given line is called the critical gas

density10, n. These rates depend on temperature as well, but more weakly than density.

Some sample values are given in Table 9.3.

From Table 9.3, the Ly a line, a recombination line which occurs in an ionized

environment, will never be collisionally de-excited under typical interstellar conditions

since the density of interstellar material (see Table 3.1) does not achieve such high

9Downwards transitions can also be radiatively induced (called stimulated emission), but typically only if theradiation field is strong. More specifically, a radiatively-induced downwards transition will occur if the timescalefor this process is shorter than the timescales for both collisionally-induced and spontaneous downwardstransitions. Although this situation is not as common in astrophysics, stimulated emission must be included undersome circumstances. See also Footnote 5 in Chapter 6.10 The term, critical density is more commonly used, but this term also has another meaning in cosmology, so wedo not employ it here.

9.2 THE LINE STRENGTHS, THERMALIZATION, AND THE CRITICAL GAS DENSITY 289

values. This does not mean that collisions are never important in an ionized hydrogen

gas, however, since other recombination lines, especially at high quantum numbers, may

indeed be collisionally-induced at lower densities, as we shall see in Sect. 9.4.1. Other

kinds of transitions in hydrogen can also be collisionally induced. For example, a

transition between two n ¼ 2 levels (2s ! 2p) becomes collisionally induced at a

density of 1:5 104 cm3 (Ref. [115]). This is higher than the density of most

interstellar ionized gas but can occur around active galactic nuclei. As for interstellar

H I clouds, these have typical densities nH>0:1 cm3, so the H I l 21 cm line, with a

critical gas density of 105 cm3, should always be thermalized. This is also the case for

the listed CO line, since most molecular clouds have densities, nH2>103 cm3.

Given the large range of possible critical densities and the number of spectral lines

that are observationally accessible (Sect. 9.1), it is sometimes possible to choose the

spectral line or lines that best probe the conditions of interest. A well-designed

observational program will consider these points before time is spent at the telescope.

We will explore how spectral lines can probe the physical conditions of the gas in Sects.

9.4 and 9.5.

9.3 Line broadening

In Appendix D.1.3, we indicated that a spectral line cannot be arbitrarily narrow. It has

a natural line width which is dictated by the time the particle spends in the upper energy

level before spontaneously de-exciting. The line shape, in the absence of other effects,

is Lorentzian, with its shape given by the line shape function, also called the line profile,

LðnÞ (Eq. D.14). Since all transitions have this quantum mechanical limitation, all

spectral lines will have a natural line width.

In real systems, however, the measured line width is always much greater than the

natural line width. This means that one or more additional line broadening mechanisms

must also be present and be sufficiently important to dominate the quantum mechanical

effects. Various line broadening mechanisms are known. In astrophysics, however, the

Table 9.3. Sample critical gas densitiesa

Species Wavelength or transition Colliding species Temperature (K) nðcm3)

H Ib Ly a (2p ! 1s) electron 104 9:2 1016

H Ic 21 cm H I 100 3 105

OIIId 493:26 nm electron 104 7:1 105

COe 2:6 mm (J ¼ 1 ! 0) H2 30 2 103

aThe critical gas density of the colliding species for the transition listed is determined by setting gj;i n ¼ Aj;i,where gj;i is the collisional rate coefficient (cm3 s1) for a transition from upper state, j, to lower state, i, and Aj;i

(s1) is the Einstein A coefficient for that transition.bUsing g from Table 3.2 and Aji from Table C.1.cUsing g ¼ 9:5 1011 (Ref. [160]) and Aji from Table C.1.dUsing data from www.astronomy.ohio-state.edu/~pradham/atomic.html with Eq. 4–11 from Ref. [160] and Afrom Ref. [160].eFrom data in Ref. [160]. J is the total angular momentum quantum number.


most important ones are Doppler broadening and pressure broadening, of which the

former is the most common and most easily linked to the physical state of the gas. In the

next two sub-sections, therefore, we describe Doppler broadening in some detail and

then discuss pressure broadening and related effects more qualitatively.

9.3.1 Doppler broadening and temperature diagnostics

Doppler broadening is due to the collective motions of the particles that are emitting

the spectral line (recall that l=l0 ¼ v=c, where v is a radial velocity, Eq. 7.3). Even if

a gas cloud (meaning any gaseous system that is emitting the line) has no net motion

towards or away from the observer (vsys ¼ 0, see Sect. 7.2.1.2) the individual particles

within the cloud will still have some motion, be it thermal, turbulent, or perhaps from

internal systematic motions such as rotation, expansion or contraction or motions

related to shock waves. If each particle within the cloud has a different motion, then the

various radial velocity components of these motions will produce different Doppler

shifts. The net result will be a superposition of all the individual Doppler shifted lines,

resulting in a broadened line. If vsys ¼ 0, then the widened line will be centred on its

rest frequency, and if vsys 6¼ 0, then the line is centred at the Doppler-shifted frequency

of the line for that vsys (Eq. (7.4), or Eq. (7.3) if plotted by wavelength). We have

already seen an example of this for a rotating galaxy (see Figure 7.4.d). When emission

from the entire galaxy is detected within a single beam, the galaxy’s rotation causes the

line profile to be spread out over a wide frequency range, or a wide velocity range if the

x axis is expressed in velocity units. Also, a rapidly expanding supernova remmant, if

unresolved, should have a line profile whose width reflects the maximum velocities of

both the advancing and receding sides of the shell. The expansion velocity is then half

of the line width (Prob. 9.4).

Not every cloud may be expanding or contracting or have other peculiar motions

associated with it, but spectral lines that are formed in a gas at some temperature, T,

will always experience at least thermal line broadening due to the Doppler shifts of

particles in a Maxwellian velocity distribution (Eq. (0.A.2), Figure 3.12). Taking a

simple case in which the cloud has no systemic velocity, an optically thin spectral line

(see below) from such a cloud has a Gaussian shape centered on the rest frequency, ni;j,

as illustrated in Figure 9.4b (lower curve).

The Gaussian line shape function (Hz1), or profile, is described by,

DðnÞ ¼1ffiffiffiffiffiffi

2 pp

sDexp ðn ni;jÞ2

2 sD2

!ð9:1Þ

where,

sD ¼ 1ffiffiffi2

p ni; j

c

2 k T

m

1=2

¼ 1ffiffiffi2

p ni; j

cb ¼ 1ffiffiffi

2p

li; j

b ð9:2Þ

9.3 LINE BROADENING 291

T is the gas kinetic temperature (K), m is the mass (g) of the particle emitting the

spectral line, b (cm s1) is the velocity parameter which is defined by Eq. (9.2), i.e.

b ¼ ð2 kT=mÞ1=2, and we have used the relation, l n ¼ c.

The full width at half maximum (FWHM) intensity, nD, is related to sD and is

given (in units of Hz) by,

nD ¼ 2:3556sD ¼ 7:14 107 ni;jT

A

1=2

¼ 2:14 104

li;j

T

A

1=2

ð9:3Þ

where we have used Eq. (9.2), set m ¼ A mH, with mH the mass (g) of the hydrogen

atom, A (unitless) is the atomic weight, and the constants have been evaluated. The

value of sD, if it is desired, can also be obtained from Eq. (9.3). Note that higher

0

T

TB

ν

τ ν > 1

τ ν < 1

νi,j

∆νD1

2

υcloud = 0

⇒Line of sight

(a) (b)

Figure 9.4. The thermal motions in a gas cloud that is globally at rest form a Gaussian-shapedspectral line, if the line is optically thin. (a) Sketch of a gas cloud at rest, showing some of theinternal particles (dots) and their velocity vectors (heavy arrows). The thin arrows show thecomponents of velocity along the line of sight that broaden the spectral line. The two labelledparticles have identical radial velocities and so will contribute to the line intensity at the samefrequency. (b) Two spectral lines plotted on a brightness temperature scale. The lower darkcurve shows the Gaussian line shape given by Eq. (9.1) with the FWHM labelled. The hatchedarea (with width exaggerated) corresponds to all particles that have equal radial velocities, likethose of particles 1 and 2 in (a). The grey higher curve shows the same line, but for a higherdensity cloud which has resulted in optically thick emission. In that case, TB ¼ T near the linecentre.


frequency lines have broader widths in frequency, which reflects the fact that

n ¼ n0 v=c from the Doppler shift. The peak of the line is,

Dðni; jÞ ¼0:9394

nDð9:4Þ

and the integral over all frequencies (a unitless quantity, like Eq. D.18) is,

Z 1

1DðnÞ dn ¼ 1 ð9:5Þ

Eq. (9.3) provides a relation between the line width in frequency, nD, which is a

straightforward quantity to measure, and the temperature of the gas, assuming that

thermal broadening is the dominant line broadening mechanism. It is sometimes more

useful to have a measure of the line width in velocity units, however, especially since

many spectra are plotted as a function of velocity, rather than frequency or wavelength

(see also Prob. 9.3). Upon applying the Doppler formula to the right hand side of

Eq. (9.3), the velocity line width is (cm s1),

vD ¼ 2:14 104 T

A

1=2

ð9:6Þ

Eq. (9.6) is now independent of the specific spectral line being observed. This must be

the case, since any given atom, which could have many transitions at different

frequencies, will move at only one velocity.

The thermal line width is larger than the natural line width (e.g. Prob. 9.3) and is

always present in a thermal gas. However, there may be other Doppler motions (e.g.

from rotation, turbulence, etc. as indicated above) which could widen the line even

more, and these often dominate the thermal motions. If turbulence is present and the

turbulent motions are also Gaussian, then it can be shown that the resulting profile, a

convolution (Appendix A.4) of the two functions, will still be Gaussian, but wider. If

other motions are present, the profile may no longer be Gaussian, as we saw for

Figure 7.4.d. Thus, for Doppler broadening, the shape and width of the line profile

provides information as to the internal velocities or temperature of the cloud.

Allowing for the possibility of additional internal motions, then, the observed

FWHM of the line, nFWHM or vFWHM, will be greater than or equal to the thermal

line width,

nFWHM nD; vFWHM vD ð9:7Þ

Together with Eqs. (9.3) or (9.6), this shows that the measured line width places an

upper limit on the temperature of the gas. This is a useful result because it means that

we need only plot the spectral line and measure its width, without doing any other


calculations, in order to determine an upper limit to T. In Sect. 6.4.1, moreover, we

noted that the peak of a spectral line in LTE provided a lower limit to the gas

temperature. Thus, provided these conditions are met, the gas temperature range can

be immediately constrained by simply plotting the spectrum and taking two relatively

straightforward measurements, as Example 9.1 describes.

Example 9.1

The l 21 cm line of hydrogen emitted from an interstellar cloud has a peak specific intensity ofIn ¼ 4:03 1017 erg s1 cm2 Hz1 sr1 and a line width of nFWHM ¼ 15:0 kHz. Find therange of possible temperatures of the cloud.

The l 21 cm line is in LTE (Sect. 9.2, Sect. 6.4.3) and therefore, by Eq. (6.24), TB T.

Since h n k T for this line (Sect. 6.4.3), we can use the Rayleigh–Jeans formula (Eq. 4.6)

to convert the specific intensity, In, at the frequency of the line centre ðvi; j ¼ 1420:4 MHzÞto a brightness temperature, finding TB ¼ 65 K for the line peak. Then using the given value

of nFWHM in Eq. (9.3) with li; j ¼ 21:106 cm (Table C.1) and A ¼ 1, we find an upper

limit to the temperature of T ¼ 219 K. We conclude that the temperature of this cloud is in

the range, 65 T 219 K.

It is worth noting that the thermal line width applies to any transition in the gas, no

matter how it is produced (e.g. collisionally, radiatively, or spontaneously) because it is

the velocity distribution of the particles that is determining this width (not the physics

of the spectral line). Thus, the upper limit to T applies to any optically thin line. On the

other hand, the condition, TB T will only be true for a line in LTE because the peak of

the line must be related to the motions of particles in the gas for this condition to have

any meaning.

We have so far considered only optically thin lines. It is also of interest to consider

what a line in LTE will look like if there is a large density of particles that contribute to

the line. For a specific transition in a given cloud, the optical depth equation (Eq. 5.8)

indicates that tn increases linearly with increasing density. Therefore, at some high

density, the line should become optically thick (tn>1). From the LTE solution to the

Equation of Radiative Transfer (Eq. 6.19), assuming no background source, then

In ¼ BnðTÞ at frequencies for which tn>1. This also means that TB ¼ T at these

frequencies (see Sect. 4.1.1). Since more particles contribute to the line centre than its

wings, the result is a flat-topped profile, as illustrated by the higher curve in

Figure 9.4.b11. The presence of flat-topped profiles, therefore, is a diagnostic for

optically thick thermalized lines. Provided we can be sure that no other effects are

11Flat-bottomed profiles also result from optically thick absorption lines. In such a case, all of the backgroundcontinuum emission is removed by the foreground line emission at and near the central line frequency. A quantitycalled the equivalent width, W , is used to measure the strength of an absorption line for any tn, i.e.W

R½ðIn0 InÞ=In0 dn, where In0 is the specific intensity of the background continuum and In is the specific

intensity measured in the line. The integral is over the line.


producing unusual profiles12 the peak brightness temperature of such a line will give

the gas temperature.

9.3.2 Pressure broadening

The second important line broadening mechanism in astrophysics is pressure broad-

ening which we consider here only qualitatively. Pressure broadening actually includes

a number of possible effects which become important when densities are higher, such

as in denser stellar atmospheres, and involve interactions between a radiating atom and

other particles around it. The interacting particles essentially perturb the radiating atom

in such a way as to cause a small shift in the frequency of an emitted spectral line. For

an ensemble of particles, each shift may be different, resulting in a widened line.

There are a number of approaches that have been taken to understand this process,

an important one being to treat the atom as a radiating harmonic oscillator, as is

done in Appendix D.1.2. Collisions (impacts) with surrounding particles then

effectively ‘interrupt’ the emitted radiation, introducing a perturbation on the phase

and amplitude of the oscillation. The collision itself does not produce a transition,

but perturbs the transition that is taking place. This interruption in time results in a

wider frequency response (recall the reciprocal relation between time and fre-

quency). The collision may be with protons, electrons, or neutral particles, and

the result is different, depending on the type of impactor. This approach has been

widely used and the line broadening is therefore called collision broadening. It can

be shown that collision broadening, for each type of impactor, leads to a Lorentzian

profile (see Eq. D.14), but with the quantum mechanical radiation damping constant,

i; j, replaced by a collisional damping constant, Pi; j. The latter is a function of

temperature, density, the specific line being considered, and the nature of the

impacting particle, and therefore the Lorentzian FWHM, nP, also depends on

these quantities (see Eq. D.17).

Since we know that thermal line broadening, which has a Gaussian profile, is always

present and its width is much greater than the natural line width, then the presence of an

observed Lorentzian profile suggests that collision broadening is occurring and is

dominating. This, then, provides information about the environment within which the

line is forming. In reality, the various line widths are difficult to model and compare

with observation, especially since collision coefficients are not known for every type of

encounter and spectral line. However, some lines have been successfully reproduced.

For example, the widths of certain strong sodium lines in the Solar atmosphere can be

adequately explained by impacts with neutral hydrogen (Prob. 9.3). If both collision

broadening and thermal Doppler broadening are important (one does not strongly

dominate the other), then both the Gaussian and Lorentzian profiles are incorporated

into the line profile. The resulting shape is called a Voigt profile.

12A spherically symmetric expanding wind which has a cut-off in radius, for example, can produce flat-toppedprofiles, even if optically thin.


An interaction may be considered an ‘impact’ if the duration of the collision is much

less than the time between collisions. However, if the impact duration approaches the

time between collisions, then the effect may be considered continuous. For example,

perturbations can occur from the collective quasi-static effects of surrounding ions.

These ions produce a net electric field which causes the energy levels of a radiating

atom to split and /or shift. Since the energy levels of many particles are no longer at one

fixed value but are spread out, the resulting spectral line will also acquire a width. This

effect is called Stark broadening because the shifts in energy levels result from

the application of an external electric field, an effect known as the Stark effect. (The

analogy for a magnetic field is the Zeeman effect, as discussed in Appendix C.3.) The

resulting line profile may no longer be Lorentzian13.

For some of the above interactions, the different approaches could be considered

different ways of describing similar effects. For example, an impact of a free electron

with a bound electron can be thought of as an encounter in which the trajectory of the

free electron changes significantly (see Footnote 22 in Chapter 3). As the free electron

encounters the atom, it briefly applies Stark broadening to the atom. Thus, line

broadening due to electron or proton collisions might also be referred to as Stark

broadening. Similarly, the net electric field produced by surrounding ions can be

thought of as the collective effects of many slow impacts. Because of this, as well

as the fact that these mechanisms all tend to be important in high pressure environ-

ments, the terms pressure broadening, collisional broadening and Stark broadening are

sometimes used without distinction in the literature, especially the first two. More

information on these mechanisms can be found in Ref. [68].

9.4 Probing physical conditions via electronic transitions

In principle, it is straightforward to write down the emission coefficient of a spectral

line that results from a spontaneous downwards transition from upper level, j, to lower

level, i,

jn ¼nj Aj;i h n

4pðnÞ ð9:8Þ

where nj (cm3) is the number density of atoms having electrons in level, j, Aj; i (s1) is

the Einstein A coefficient, h is Planck’s constant, n (Hz) is the frequency of the line, and

ðnÞ (Hz1) is the line shape function describing the relevant line broadening mechan-

ism (e.g. Eq. 9.1). A dimensional analysis of this equation will verify that the units are

erg s1 cm3 Hz1 sr1, as required for jn (Sect. 6.2).

In practise, however, it is often a challenge to arrive at a full description of the

intensity of a spectral line. For example, the population of the upper state, nj, must be

13Stark broadening can be of two types, called linear and quadratic, depending on the strength of the frequencyshift with particle separation. The line profile shape is different for the two types.


found, and collisional and radiative de-excitations, as well as ionization equilibrium

must also be included, if important. The full equations of statistical equilibrium may

therefore be required (see Sect. 3.4.4) in order to compute emission coefficients for the

various lines in the atom. To compute the observed spectral line intensities, In, the

emission coefficient must be used in the appropriate solution to the Equation of

Radiative Transfer (Sect. 6.3) which requires that accurate optical depths be deter-

mined. The optical depths will vary from line to line and will also vary, according to

some line profile shape, within each line. The type and extent of the line broadening

mechanism is also important. For example, a line that would normally be optically

thick could be optically thin if the cloud is undergoing large-scale turbulence or other

motions that spread out the line and lower the line peak. This, in turn, affects the

radiative transfer through the cloud. Finally, the spectral lines may sit atop an under-

lying continuum (e.g. Figure 8.9).

Fortunately, however, some conditions exist for which simplifications can be made

without badly compromising the results. We will now look at some of these and

consider what the observed spectral lines can tell us about the physical state of the

gas. We will start with high quantum numbers and work ‘down the quantum number

ladder’, focussing on three main transitions or types of transitions: radio recombination

lines, optical recombination lines, and the l 21 cm line of H I. Recombination lines will

be seen in any ionized gas, including H II regions, planetary nebulae, and diffuse

ionized gas in the interstellar medium. The l 21 cm line will be seen in neutral H I

clouds (Sect. 3.4.7). As before, we will focus mainly on the hydrogen atom, but there

are many other constituents besides hydrogen that emit their own spectral lines, each

providing further diagnostics of the physical state of the gas.

9.4.1 Radio recombination lines

At high quantum numbers (e.g. n040), the energy levels of the hydrogen atom become

very close together (see Figure C.1), so transitions between these levels result in

radiation at radio wavelengths. These transitions represent the upper quantum number

limit to the recombination line spectrum (designated Hna for a transition from

principal quantum number n þ 1 to n, Hnb for n þ 2 to n, etc., see Appendix C.1)

and are called radio recombination lines (RRLs). At high n, hydrogen transition

probabilities are low, so RRLs tend to be weak in comparison to lower quantum

number optical recombination lines (Sect. 9.4.2). Nevertheless, many such observa-

tions have been made, not only of hydrogen RRLs, but of many other species as well

(see Figure 9.5). RRLs from quantum numbers as high as n ¼ 766 have been measured

from carbon, for example, corresponding to frequencies as low as 14 MHz. For

hydrogen, RRLs have so far been observed at lower n but, in principle, it is believed

that quantum numbers up to n 1600 are possible14 (Ref. [66]). Using Eq. (C.3), the

14Such high quantum number transitions are more likely to occur in gas that is very low density, cooler, and withlittle background radiation. See Ref. [66] for further information.

9.4 PROBING PHYSICAL CONDITIONS VIA ELECTRONIC TRANSITIONS 297

diameter of such an atom would be an astonishing 0:3 mm! This is larger than the

thickness of a human hair, and is three million times larger than the size of the hydrogen

atom in its ground state.

As previously indicated (see Sect. 8.2.2), radio waves are not hindered by dust and can

therefore can be seen over great distances. If the optical emission of an H II region is

obscured by extinction, radio (or long wavelength infrared) observations may be the

only way of detecting that an H II or other ionized region is present at all (see Figure 9.6).

Any ISM component that suffers from heavy extinction can be similarly probed at radio

wavelengths. Ultra-compact (UC) H II regions, for example, represent a class of H II

regions which are almost always heavily obscured, optically. These are small (90:1 pc),

very dense (0104 cm3) H II regions around O or B stars that are still embedded within

their natal dusty molecular clouds (recall Sect. 5.2.2, see also Ref. [42]). They therefore

represent a relatively early stage in the development of massive star forming regions and

long wavelength data are required to probe their characteristics. RRLs also provide an

Figure 9.5. The radio recombination line spectrum of the H II region, Sharpless 88 B, near n ¼ 5GHz using the Arecibo telescope (see Figure 2.6a). The region that has been observed is circular onthe sky of angular diameter, yb ¼ 59 arcsec. The continuum has already been subtracted. Notice thepresence of various weaker recombination lines, besides the H109 a line of hydrogen. (Reproducedby permission of Yervant Terzian)


opportunity to probe the diffuse ionized gas (DIG, Sect. 8.2.2) over large distances in the

Galaxy (e.g. Ref. [40]). Together with a model of galaxy rotation, RRL Doppler shifts

can be used to obtain distances to the line-emitting gas (Sect. 7.2.1.2), something that

cannot be achieved by continuum measurements alone.

Figure 9.6. The massive star forming region, G24.8+0.1, located in the plane of the Milky Way(see Figure 3.3), is estimated to be at a distance of D ¼ 6:5 kpc from us and would be completelyinvisible were it not for observations at long wavelengths. (a) This optical image shows only nearbystars since the optical emission from G24.8+0.1, itself, is heavily obscured by foreground dust. (b)This image, taken at n ¼ 1281:5 MHz using the Giant Metrewave Radio Telescope (GMRT) in India,shows the same region as in (a). Here we see complex radio continuum emission containing anumber of HII regions. Most of the emission is shell-like, but a reasonable assumption is that itoccupies a volume which is equivalent to a sphere of diameter, 40. The radio continuum flux comingfrom this volume is f1282 MHz ¼ 7:7 Jy. (c) The H172 a recombination line, at the same frequency,from the region shown in (b) is displayed in this image. The underlying continuum has already beensubstracted from this spectrum. LSR refers to Local standard of Rest (see Sect. 7.2.1.1). (GMRTimages Reproduced by permission of N. Kantharia)


For RRLs, several assumptions can usually be made that simplify the analysis of

these lines. Firstly, RRLs are weak lines and are observed in the optically thin limit.

Secondly, at high quantum numbers, the spontaneous downwards transition rate is a

very strong function of quantum number, i.e. Anþ1!n / n6 (Ref. [50]), so the time-

scale for spontaneous de-excitation becomes very large (e.g. see the H109a line value

in Table C.1 for which t 103 s). This means that RRLs tend toward LTE conditions

(Sect. 9.2)15. And thirdly, at radio wavelengths, h n k T so the Rayleigh–Jeans

approximation to the Planck function (Eq. 4.6) can be used. Thus (as was the case

for thermal Bremsstrahlung emission observed at radio wavelengths, see Sect. 8.2.2)

we can use Eq. (6.33) for the line, TBn ¼ Te tn where, in this case, tn will be a function

of the line shape, among other parameters.

We now assume that the line shape is Gaussian (applicable to thermal broadening

alone or thermal broadening with Gaussian turbulence, see Sect. 9.3.1) in order to write

an expression for the optical depth at the centre of the line, tL. For transitions between

quantum levels, n, and n þn, this is given by (Ref. [55]),

tL ¼ 1:01 104 Z2 nfn;nþn

n

Te

2:5 ewn

n2 k Te

n1EML

ð9:9Þ

where Z is the atomic number, fn;nþn is the oscillator strength for transition,

n ! n þn (see Appendix D.1.3), wn (erg) is the ionization potential (Sect. 5.2.2)

of level n, n (kHz) is the FWHM of the Gaussian line, and EML (cm6 pc) is the

emission measure of the line given by Eq. (8.5), where it is assumed that the ion and

electron densities are roughly equal. Taking n ¼ 1, Z ¼ 1, ðfn;nþ1Þ=n 0:1908 for

large n when n ¼ 1 (Ref. [177]), and noting that the value of the exponential

approaches 1 for large n, then Eq. (9.9) becomes, for the centre of a Hna RRL,

tL ¼ 1:92 103 Te

K

2:5 nkHz

1 EML

cm6 pc

ð9:10Þ

The brightness temperature of the centre of an optically thin line can then be written

(Eq. 6.33 and above),

TL ¼ 1:92 103 Te

K

1:5 nkHz

1 EML

cm6 pc

ð9:11Þ

As we have seen before, emission from an optically thin source is a function of both the

temperature and density, the latter via the emission measure. Thus, an observation of a

single RRL cannot lead to information on one of these quantities without knowing the

other.

15Typically, if n>168 ne, where ne is the electron density, LTE can be assumed (Ref. [50]). However, stimulatedemission (see footnote 5 in Chapter 6) can also affect the line strength. See Ref. [137] for information as to whenstimulated emission and non-LTE conditions may apply.


There is additional information associated with RRL observations, however, that

aids our efforts in finding both the temperature and density of the gas. The same

ionized gas that produces RRLs will also emit continuum radiation. At radio

wavelengths, this continuum is due to thermal Bremsstrahlung (Sect. 8.2) for which

the optical depth and brightness temperature were given by Eqs. (8.12) and (8.13),

respectively. Thus, the RRL sits on top of continuum emission16 and TL is a value

that is measured from the level of the continuum ‘upwards’. Usually, the continuum

emission is first subtracted from the total lineþcontinuum before the measurement is

made (e.g. Figure 9.5). If the continuum emission is also optically thin17, then the

line-to-continuum brightness temperature ratio at the line centre, r, can be formed by

dividing Eq. (9.11) by Eq. (8.13) or, equivalently, dividing Eq. (9.10) by Eq. (8.12).

The result is (Ref. [138]),

r ¼ TL

TC

¼ tL

tC

¼ 2:33 104 Te

K

1:15 nkHz

1 nGHz

h i2:1

ð9:12Þ

where the subscripts, L and C, refer to the line and continuum, respectively.

In practise, we cannot measure the value of the continuum at exactly the line centre

(e.g. Figure 6.3b), but we can measure it just adjacent to the line, as was done for the

l 21 cm line in Sect. 6.4.3. In Eq. (9.12), we have assumed that EML=EMC ¼ 1 which

would be the case for a pure hydrogen ionized gas18. Thus, the line-to-continuum ratio

can be used to find the electron temperature of the ionized gas explicitly. This is a very

useful result because, if the assumptions are correct, a determination of Te requires

only a measurement of a single line together with its continuum, and the result is

independent of distance. The electron density can then be found from either the line or

continuum measurement, provided the distance is known, as Example 9.2 outlines.

Example 9.2

The H93 a recombination line has been measured (Ref. [2] ) within a 500 500 square regioncentred on the ultra-compact H II region, G45.12 (D = 6:4 kpc). The observing frequency isn ¼ 8:044 GHz, the line FWHM is n ¼ 1497 kHz, and the flux density in the continuum andline centre are fC ¼ 1:72 Jy and fL ¼ 0:059 Jy, respectively. Assume that the line andcontinuum, both optically thin, are formed in the same physical region and find Te, tL, tC,EM, and ne of this region.

16The condition for emission lines which requires that the line-emitting gas be hotter than the backgroundcontinuum (see Sect. 6.4.2) does not apply in this case because there is no ‘background’; both the line andcontinuum come from the same ‘pocket’ of gas.17At very low frequencies, corresponding to very high quantum numbers, the line will be optically thin but thecontinuum may be optically thick (see Eq. 8.12).18As noted in Footnote 7 of Chapter 8, singly-ionized helium will contribute free electrons which will increase thecontinuum emission, but not the line emission, as derived. To account for ionized helium, the right hand side ofEq. (9.12) should by multiplied by the factor, 1=½1 þ NðHeIIÞ=NðH IIÞ.


From the given values, fL=fC ¼ 0:059=1:72 ¼ 0:0343, but using Eqs. (1.13), (4.4), and

(4.6), fL=fC ¼ TL=TC, where TL and TC are the brightness temperatures of the line and

continuum, respectively. We called this ratio, r, in Eq. (9.12), so r ¼ 0:0343. Using the

given values of n and n, Eq. (9.12) allows us to find the electron temperature, resulting in,

Te ¼ 9198 K.

The angular size of the observed 500 500 square region is ¼ 5:876 1010 sr, upon

converting units. At a distance of D ¼ 6:4 kpc, the angular diameter corresponds to (Eq.

B.1) l ¼ 0:155 pc which we also take to be the line of sight distance through the source.

Thus, the geometry is that of a ‘box’ for this example. The optical depths, tL and tC, can be

found from TBn ¼ Te tn (Eq. 6.33) which requires that individual brightness temperatures

be computed (only the ratio is known, from above). Converting the flux densities to specific

intensities (Eqs. 1.13 and 1.8), yields IL ¼ 1:00 1015 erg s1 cm2 Hz1 sr1 and

IC ¼ 2:93 1014 erg s1 cm2 Hz1 sr1 for the line and continuum, respectively.

Converting from specific intensities to brightness temperatures (Eqs. 4.4 and 4.6) gives

TL ¼ 50:3 K and TC ¼ 1475 K for the line and continuum, respectively. With the known

value of Te, this gives tL ¼ 0:0055 and tC ¼ 0:16.

To compute the EM, either the line (Eqs. 9.10 or 9.11) or continuum emission (Eqs. 8.12

or 8.13) can be used19, both of which lead to EM ¼ 34:7 106 cm6 pc. We now use Eq.

(8.5) with the value of l from above to obtain ne ¼ 1:5 104 cm3.

Note that the calculated values are averages that are representative of the region as a

whole. The density, for example, likely varies considerably within the region. A more

accurate calculation for this source, which includes non-LTE effects, leads to Te ¼ 8300 K

and ne ¼ 1:8 104 cm3 (Ref. [2]), i.e. errors of order 10 per cent in temperature and a

15 per cent in density have resulted from an LTE assumption.

As Example 9.2 has pointed out, some error is introduced into the calculations by

making the assumptions outlined in this section. The magnitude of this error will

vary, depending on the RRL adopted and the physical conditions in the gas (see Ref.

[153] for examples). Thus, Eq. (9.12) can be used to obtain an initial estimate of the

electron temperature, but a full non-LTE approach will yield more accurate results

(Prob. 9.6).

It is interesting that, at very low frequencies, some RRLs can be seen in absorption

against the brighter Galactic background, the latter due primarily to synchrotron

radiation that is stronger at lower frequencies (see Eqs. 8.42, 8.43). As an example,

the absorption line of singly ionized carbon, C575 a has been observed at n ¼ 32:4MHz (Ref. [83]).

9.4.2 Optical recombination lines

As we step down the ‘ladder’ of quantum numbers, we eventually reach the more

familiar Balmer lines of hydrogen which are observed in the optical part of the

19Eq. (8.17) is not applicable for this example since it uses a different geometry from what has been assumed.


spectrum (see Table C.1). Given the lengthy history of optical observations in compar-

ison to any other waveband (Sect. 2.1), it is not surprising that images in the Balmer

lines, especially the red H a line, are plentiful. Just as we saw for the RRLs, optical

recombination lines are formed in ionized gas, and many beautiful and striking images

of H II regions, planetary nebulae, and other ionized regions have been obtained in the

H a line as well as in emission lines from other species. Figures 3.7, 3.13, 5.10, and 6.6

provide some examples. The spectrum of an H II region was also shown in Figure 8.9,

illustrating the rich abundance of lines available for study. What can these lines tell us

about the physical conditions in the ionized gas?

For hydrogen in the relatively low density interstellar medium, optical recombina-

tion lines are not in LTE. The lifetimes of electrons in their various states are very short

(see the Einstein A coefficients in Table C.1) and downwards transitions will be

spontaneous. This means that the line emission coefficient, jn, can be expressed by

Eq. (9.8). For optically thin lines, then, in the absence of a background source, the

specific intensity of a line simply requires a multiplication of jn by the line-of-sight

depth through the cloud, l, as indicated by Eq. (6.17). The problem, then, reduces to one

of finding the number of particles in the upper state, nj, that can be used in Eq. (9.8).

This task is less trivial and requires a consideration of the recombination of free

electrons with protons, non-LTE conditions, and quantum mechanical selection rules

as electrons cascade down various levels in the atom. The result, integrated over the

line (erg s1 cm3 sr1), is,

Zn

jn dn ¼ 1

4 ph nj;i aj;i ne ni

1

4 ph nj;i aj;i ne

2 ð9:13Þ

where nj;i is the frequency of the line centre, aj;i is the effective recombination

coefficient for transition, j ! i (cm3 s1), ni, ne are the ion and electron densities,

respectively, and the latter approximation applies if the ion and electron densities are

equal which we take to be the case. The recombination coefficient has appeared before

when we considered the ionization equilibrium of an H II region (see Example 5.4) for

which ar included recombinations to all levels of the atom. In the above case, aj;i will be

different for different transitions and is defined such that aj;i ni ne is the total number of

photons per second per cubit centimetre in transitions from level j to i. Since recombi-

nation is a collisional process, the effective recombination coefficients, as we saw for ar

(see Table 3.2), are functions of temperature. Several values are provided in Table 9.4.

We can now use Eq. (9.13) with Eq. (6.17) to obtain the intensity (erg s1 cm2 sr1)

over the line, assuming ni ne,

Zn

In dn ¼ 1

4 ph nj;i aj;i n2

e l ¼ 2:46 1017 h nj;i aj;i EM ð9:14Þ

where we have expressed the emission measure, EM (see Eq. 8.5) in the usual units of

cm6 pc. As usual for thermal gas in the optically thin limit, the emission is a function

of temperature (via aji) and density (via EM).


We now come to an interesting point. The effective recombination coefficients are

well known and have been tabulated for a wide range of conditions. Also, for any two

lines in a given atom, the emission measure should be the same (see also Sect. 9.4.1).

This means that the ratio of the intensities of any two lines should be a function of

frequency and the effective recombination coefficient, only. Forming this ratio from

Eq. (9.14).

Rn In dnðj ! iÞRn In dnðk ! mÞ ¼

nj;i aj;i

nk;m ak;mð9:15Þ

where j ! i and k ! m represent the two different line transitions.

Thus, the various line ratios can be calculated and tabulated as a function of

temperature, since aj;i is a function of temperature. In principle, then, measuring the

line ratio of any two such lines can provide us with the temperature by comparison with

the tabulated values. Since the ratio is formed from the integrals over two lines, the

result is not dependent on the type of line broadening that is present, as was the case for

RRLs for which a line/continuum ratio was formed. Thus, we have yet another

temperature diagnostic for H II regions or other interstellar ionized gas clouds. The

density could then be found via Eq. (9.14) with a knowledge of Te and distance to

the object. In practice, as we will now describe, there are a few complications to this

simple approach using the Balmer lines of hydrogen that need to be considered.

If all hydrogen lines were optically thin (a situation called Case A), then there should

be no density dependence to aj;i or to the line ratios of Eq. (9.15). However, as pointed

out in Sect. 5.1.1.2, the Lyman lines of hydrogen, which emit in the UV, tend to be

trapped in ionized nebulae (a more realistic and density-dependent condition called

Case B). For example, after several scatterings, there is some probability that a Ly bline (a transition from n ¼ 3 to n ¼ 1) will be degraded into an H a (n ¼ 3 to n ¼ 2) and

then a Ly a (n ¼ 2 to n ¼ 1) line. Even though the Balmer lines themselves are

optically thin, the Balmer line ratios will still be affected. Thus, the values of aj;i given

in Table 9.4 are for Case B and a sample line ratio calculation for this case is provided

in Example 9.3. Expected line ratios are tabulated for this case in Table 9.5. The

corrections are not large. Over the range of temperature and density listed,

the maximum difference between Case A and Case B is only 3 per cent and, as can

be seen in the table, the maximum error introduced by changing the density a factor of

100 is even less.

Table 9.4. Effective recombination coefficientsa

Temperature (K)

aj;i (cm3 s1) 5 000 10 000 20 000

a3;2 (H a) 2:21 1013 1:17 1013 5:97 1014

a4;2 (H b) 5:41 1014 3:03 1014 1:61 1014

aCase B recombination has been assumed (see Sect. 9.4.2). H a values are from Ref. [160] and Hb values arefrom Ref. [115]. For the latter reference, means for the densities, 102 and 104 cm3 are given.


Example 9.3

Compute the expected line ratio, H a/H b, for an H II region of temperature, 104 K.

From Eq. (9.15), we can write,

H aH b

¼ n3;2

n4;2

a3;2

a4;2¼ l4;2

l3;2

a3;2

a4;2¼ 486:132

656:280

1:17 1013

3:03 1014

¼ 2:86 ð9:16Þ

where we have used c ¼ l n and data from Tables 9.4 and C.1.

Of greater concern is that the dependence of the Balmer line ratios on tempera-

ture is actually rather weak. Looking at Table 9.5, a change in temperature of a

factor of two results in changes in line ratios of only a few per cent – in other

words, about the same as changing the assumption from Case A to Case B. The

line ratios as well as model calculations for aj;i would have to be measured and

calculated, respectively, to very high accuracy for these ratios to provide useful

temperature diagnostics.

The solution is to use other line ratios that show stronger temperature dependences

and for which all lines are optically thin. Fortunately, such lines do exist, the most

commonly used ones being associated with [OIII] and [NII]. The square brackets

refer to the fact that the relevant lines are forbidden (e.g. see Appendix C.4).

For forbidden lines, the lifetime in the upper state is relatively long yet, in interstellar

space, the densities are still low enough that emission results from spontaneous

transitions. These lines were first discovered in nebulae and the substance responsible

for them was originally named ‘nebulium’ since the same lines could not be seen

in the higher density gases of Earth-based laboratories in which collisionally-induced

transitions dominated. The unknown lines were later identified as being due

to forbidden [OIII] and [NII] transitions. It is useful to form ratios using three

Table 9.5. Sample line ratios for the hydrogen balmer seriesa

Te (K)

5 000 10 000 20 000

ne (cm3) ne (cm3) ne (cm3)

Line ratio 102 104 102 104 102 104

H a/Hb 3:04 3:00 2:86 2:85 2:75 2:74

H g/Hb 0:458 0:460 0:468 0:469 0:475 0:476

H d/Hb 0:251 0:253 0:259 0:260 0:264 0:264

H E/H b 0:154 0:155 0:159 0:159 0:162 0:163

aRef. [50]. Case B recombination has been assumed (see Sect. 9.4.2).


lines for each species, designated RO and RN for [OIII] and [NII], respectively (Ref.

[96]),

RO ¼ Iðl 4959Þ þ Iðl 5007ÞIðl 4363Þ ¼ 7:73 exp½ð3:29 104Þ=Te

1 þ 4:5 104 ½ne=ðTeÞ1=2ð½OIIIÞ ð9:17Þ

RN ¼ Iðl 6548Þ þ Iðl 6583ÞIðl 5775Þ ¼ 6:91 exp½ð2:50 104Þ=Te

1 þ 2:5 103 ½ne=ðTeÞ1=2ð½NIIÞ ð9:18Þ

where the ratio is most sensitive to temperature in comparison to density (Prob. 9.7).

These are not the only optical diagnostics of ionized regions. The line-to-continuum

ratio can also be used, as was done for RRLs (though this ratio involves more

complexity given the different processes that contribute to the optical continuum),

and other line ratios, for example, from [SII] and [OII], are particularly sensitive to

electron density, ne. Thus, by carefully choosing the appropriate lines, a variety of

optical observations can successfuly probe the physical conditions in H II regions,

planetary nebulae, and the diffuse ionized interstellar gas.

There remains one more complication that must be considered when dealing with

optical observations – that of obscuration by dust. Even if the spectral lines are

optically thin, there will still be dust throughout ionized regions, as illustrated in

Figures 3.13 and 8.4. If the extinction (Sect. 3.5.1) were independent of wavelength,

then in forming a line ratio, the effect would be the same for both lines and the ratio

would not be affected. However, extinction is wavelength dependent. Optical transi-

tions that fall into the blue part of the spectrum will be affected by dust more than

transitions in the red part of the spectrum, hence the term ‘reddening’ to describe this

effect. As Figure 3.21 showed, ElV / 1=l, roughly, in the optical part of the

spectrum, but there can be variations depending on location, so a ‘location-specific’

correction is desirable.

One way to deal with this problem is to use lines, if possible, that are very close

together in frequency (for example, the [OII] l 3727=l 3726 ratio) so that extinction is

approximately the same for the lines. Another approach, however, is to use line ratios to

correct for the reddening. Since the Balmer lines, for example, are not strongly

dependent on temperature and density, significant departures from the expected ratios

of Table 9.5 are more likely to be due to reddening than to intrinsic variations in nebular

properties. Corrected intensities would then be used in equations such as Eq. (9.17) or

(9.18).

Thus, the rich spectra seen in the optical waveband yield the properties of the

nebulae, from information on the elements that they contain, to their densities and

temperatures, and even their dust content. These are all bits and pieces of a broader

mosaic to which we can appeal in an attempt to form a coherent picture of star

formation and evolution, such as described in Chapter 3.


9.4.3 The 21 cm line of hydrogen

In 1944, the Dutch astronomer, Jan Oort, turned to a young H. C. van de Hulst and said,

‘‘. . . by the way, radio astronomy can really become very important if there were at

least one line in the radio spectrum.’’ (Ref. [186]). Within the year, van de Hulst had

predicted the presence of the l 21 cm line of H I and, by 1951, three independent teams

(Dutch, Australian, and American) successfully detected this line, each publishing

their results in the same issue of the scientific journal, Nature.

The l 21 cm line originates from the bottom rung of the hydrogen atom’s ladder.

Resulting from a tiny electron that flips its spin, the corresponding transition is between

two hyperfine levels of the ground state (see Appendix C.4). Not being hindered by

dust, this radio emission can be seen clear across the Galaxy or, for that matter, through

any other galaxy or intergalactic space. Oort’s prediction was prophetic. The impor-

tance of this apparently insignificant quantum event cannot be overstated. It is largely

through observations in the l 21 cm line that we know about the frothy nature of the

neutral interstellar medium (e.g. Figures 3.17, 6.10), the large-scale structure and

rotation of the Milky Way (see discussion in Sect. 7.2.1.2), the structure of the ISM in

other galaxies, the rotations of other galaxies (e.g. Figure 2.20) and implied presence of

associated dark matter (Sect. 3.2), tidal interactions between galaxies (see Figure 9.7,

and the expansion of the Universe itself (Sect. 7.2.2). Here, we concentrate on relating

the observed l 21 cm line to the mass of H I.

We have already seen that essentially all hydrogen atoms in a neutral cloud will have

their electrons in the ground state (Sect. 3.4.5). We have also seen how the temperature

of H I gas can be determined via the absorption and emission of the l 21 cm line when a

Figure 9.7. (a) Optical image of the M 81 Group (distance, D ¼ 3:6 Mpc). (b) This HI imageclearly shows that the three galaxies are interacting, with tidal bridges between the galaxies andother intergalactic neutral hydrogen gas. For M 81, the average column density is N HI 4:0 1020

cm2 within an elliptical region of semi-major semi-minor axes, 20 15 arcmin. (c) Thisnumerical simulation has reproduced the interaction between the galaxies. The gas originallyassociated with each galaxy is represented by a different colour. Refs. [184], [185]. (Reproduced bypermission of M. Yun). (See colour plate)


background source is present (Sect. 6.4.3). Since this is a forbidden line, the lifetime in

the upper state is long and, under typical astrophysical conditions, the line is colli-

sionally excited and therefore in LTE (Sect. 6.4.3). This means that the Boltzmann

Equation (Eq. 3.23) can be used to describe the relative populations of the two

hyperfine levels. Since h n k T for this line, the Boltzmann factor becomes 1 so

the Boltzmann Equation gives an expression for the population of states that is simply

the ratio of their statistical weights.

N2

N1

¼ n2

n1

¼ g2

g1

¼ 3

1ð9:19Þ

where the subscripts, 2, and 1, refer to the upper and lower hyperfine energy levels,

respectively and we have also expressed this ratio in terms of the ratio of number

densities, n2=n1. Therefore, the number density fraction of all particles in the upper and

lower states, respectively are,

n2

nH I

¼ 3

4

n1

nH I

¼ 1

4ð9:20Þ

where nH I is the total number density of neutral particles.

The optical depth is then (Eqs. 5.8, 5.9),

tn ¼ sn

Zl

n1 dl ¼ 1

4sn

Zl

nH I dl ¼ 1

4sn N H ð9:21Þ

where sn is the effective cross-section (cm2) (presumed not to vary along the line of

sight) dl is an element of line-of-sight distance through the cloud (cm), and the column

density, N H I (cm2) of hydrogen is defined by,

N H I Z

l

nH I dl ð9:22Þ

Thus, the column density of some species is the number density of that species

integrated over the line of sight.

For the l 21 cm line, it can be shown that the effective cross-section is given by (Ref.

[138]),

sn ¼h c2

8 p n1;2 k

g2

g1

A2;1n

TS

¼ 1:04 1014 n

TS

ð9:23Þ

where n1;2 is the frequency of the transition, A2;1 is the Einstein A coefficient for the

transition (see Table C.1), n is the line shape function (Eq. 9.1 being one example),


and TS is the spin temperature (see Sect. 3.4.5) of the gas, presumed to equal the kinetic

temperature for a line in LTE.

We can now put Eq. (9.25) into Eq. (9.21) and integrate over the line,

Zntn dn ¼ 1

4ð1:04 1014Þ N H I

TS

Znn dn ð9:24Þ

By definition, the integral over the line shape function is 1 (see Eqs. 9.5 and D.18, for

example), so Eq. (9.24) can be rearranged to solve for the H I column density,

N H I ¼ 3:85 1014 TS

Zntn dn ð9:25Þ

TS is presumed to be constant along a line of sight20 but no such assumption is

necessary for the density. When the l 21 cm line spectrum is plotted, the Doppler

relation (Eq. 7.4) is often applied so that velocity, rather than frequency, is along the x

axis (e.g. Figure 6.3). In such a case, Eq. (9.25) becomes,

N H I ¼ ð1:82 1018ÞTS

Zv

tv dv ð9:26Þ

where N H I is still in cgs units of cm2 but the velocity is in km s1. We now use

Eq. (6.31) to substitute for tv,

N H I ¼ ð1:82 1018ÞTS

Zv

ln 1 TBðvÞTS

dv ð9:27Þ

Notice that TBðvÞ is an observed quantity, so the column density can now be

found, provided there is some knowledge of TS. Also note that, since TB TS

(Eq. 6.24), the logarithmic term will be negative, so the column density is positive,

as required.

As we have done before, let us consider the optically thick and optically thin limits of

Eq. (9.27). If the line is optically thick at some velocity, v, then TBðvÞ ¼ TS (Eq. 6.32),

and the logarithm cannot be evaluated. In such a case, the column density of the cloud

cannot be obtained because we can only see into the front of the cloud (see Figure 4.2).

On the other hand, if the line is optically thin, then Eq. (6.33) can be substituted into

20The variation in temperature within some region of the ISM is usually less than the variation in density.However, should the spin temperature vary significantly along the line of sight, then the appropriate temperatureto be used in these equations is a harmonic mean temperature, weighted by HI gas density, n. More explicitly,

1TSðvÞ

D E¼R

lnv ðlÞTSðlÞ dl

h i=R

lnvðlÞ dl

, where l refers to position along the line of sight and v is the velocity (linked

to frequency via the Doppler formula, Eq. 7.4) of the gas. See Ref. [138] for further details.


Eq. (9.26) to find,

N H I

cm2

¼ ð1:82 1018Þ

Zv

TBðvÞK

dv

km s1

ðtv 1Þ ð9:28Þ

where we explicitly indicate the units. Now the column density depends only on the

observed brightness temperature integrated over the line, without the need to know the

gas temperature. It is also independent of distance. This is an important result and is

used widely since l 21 cm lines are very often optically thin. Once the column density

is known, the volume density and/or H I mass can be found, provided the distance is

known, as Example 9.4 indicates.

Example 9.4

The H I line emission from the galaxy, NGC 2903, is optically thin and subtends a solid angle ofs ¼ 1:8 105 sr. This emission, averaged over s , and then integrated over all linevelocities, is

RIv dv ¼ 0:0458 Jy beam1 km s1 (Figure 2.20 shows how this emission is

distributed in the data cube), where the beam solid angle is b ¼ 5:21 109 sr. Usinginformation given in Figure 2.11 and the solution to Prob. 7.5, find the H I mass of NGC 2903and what fraction of the total mass this represents.

To use Eq. (9.28), we require the integral over the line to be in units of K km s1. From the

definition of the Jy (Eq. 1.8) and the beam solid angle, b, we find,R

In dv ¼ 8:79 1017

erg s1 cm2 Hz1 sr1 km s1. Then using the Rayleigh–Jeans relation (Eq. 4.6) to

convert to a brightness temperature at the line centre frequency, we have,R

vTB dv ¼ 142 K

km s1 as an average over s. Putting this into Eq. (9.28) yields an average column density

of N H I ¼ 2:58 1020 cm2.

Now if this column density is multiplied by the area of the source, the result will be the

total number of neutral hydrogen atoms in the galaxy. The area is given by ss ¼ r 2 s (Eq.

B.2), where r is the distance to the galaxy (r ¼ 8:6 Mpc, see Figure 2.11), so

ss ¼ 1:27 1046 cm2. Therefore, the total number of H I atoms is N ¼ N H I ss ¼3:27 1066. Multiplying this by the mass of an H I atom and dividing by the mass of

the Sun (Tables G.2, G.3), yields MH I ¼ 2:7 109 M.

From Prob. 7.5, we found a total mass for this galaxy of Mtot 1:6 1011 M. There-

fore, the H I mass constitutes about 2 per cent of the total mass.

A number of the steps in Example 9.4 can be folded into a single equation for the H I

mass. The result is,

MH I

M

¼ 2:35 105 D

Mpc

2 Zv

fv dv

Jy km s1

ð9:29Þ

where fv is the total flux density of the source at some velocity in the line and D is the

distance to the source, in the specified units. Eq. (9.29) is commonly used to obtain the

H I mass of an object when the flux density is known at each velocity in the line.


9.5 Probing physical conditions via molecular transitions

Stars are formed from dense, dusty molecular clouds in the interstellar medium and, as

Figure 9.2 showed, many molecular lines can be excited in the warm, dense molecular

envelopes around new-born stars or in the nearby environment. Farther from local sites

of star formation, however, are extended molecular clouds that permeate the disk of the

Milky Way. These may also include embedded star formation, but the extended clouds

(for example, giant molecular clouds, or GMCs, which may be 100 pc in size) have

conditions that are less amenable to the excitation of so many lines. Typical tempera-

tures and densities are T ¼ 20 K and n ¼ 103, both of which are considerably lower

than the highest temperatures and densities found in the Orion KL cloud (Sect. 9.1.2).

Nevertheless, many lines are still accessible and provide probes of the molecular

conditions, especially rotational transitions which do not require as much energy to

be excited as do vibrational ones.

Hydrogen is the most abundant element in the Universe and it is not surprising

the most abundant molecule, by far, is H221. Determining the mass of H2 gas

is equivalent to finding the mass of the molecular cloud, since the quantities of

all other molecules and dust are much less. However, because H2 has no

permanent electric dipole moment, it cannot emit dipole radiation from rotational

transitions (Sect. 9.1.2). Moreover, even the much weaker quadrupole transitions

require relatively high energies to excite. For example, the first rotational state

above the ground state for H2 requires temperatures of 510 K which is much

higher than in the deep interstellar medium. The conclusion is that the most

abundant molecule cannot be observed in the state in which it is most commonly

found.

How then can we observe interstellar molecular clouds? The alternative is to

consider the next most abundant molecule, which is carbon monoxide.

9.5.1 The CO molecule

The CO molecule has an electric dipole moment (Sect. 9.1.2) and, as indicated in

Table 9.3, the critical density of the CO(J ¼ 1 ! 0) rotational line at l 2:6 mm, at a

molecular cloud temperature of 30 K is approximately the same as the typical density

that is observed in such clouds. This implies that this rotational CO line is collisionally

excited, and the species with which it collides will be the most abundant molecule, H2.

The l 2:6 mm line is a strong line, is easily observed, and in fact, is often optically

thick. Other CO lines may also be present, depending on excitation conditions (for

example, there are lines at l 1:3 mm, and l 0:87 mm), and lines of the C18O (e.g.

Figure 3.20) and 13CO isotopes, which are weaker and optically thin, may also be

21The process of formation of this and other molecules is still uncertain. The current view is that molecules areformed in the presence of a catalyst, probably on the surfaces of dust grains, after which they evaporate into thegas phase again.

9.5 PROBING PHYSICAL CONDITIONS VIA MOLECULAR TRANSITIONS 311

observable. Therefore, a variety of CO lines provide important diagnostics of inter-

stellar molecular clouds. Moreover, the dynamics of the cloud can also be accessed via

the Doppler shifts of these lines.

Since the density of the molecular cloud is essentially the density of its constituent

H2 molecules, much work has been expended in an attempt to determine the density of

H2 that is required to excite the various CO lines to their observed values, especially the

CO(J ¼ 1 ! 0) line, which is most readily observed22. This process is not straightfor-

ward because, unlike the l 21 cm line for which the quantity of H I can be determined

directly from a line of the same atom, the observed CO lines are used to infer the

quantity of the colliding species, H2. For example, non-LTE conditions may apply and

optical depth effects are important. The weaker, optically thin isotopic lines may be

used, but they are not always observed and isotopic ratios may not be as well known as

desired. Moreover, unknown amounts of carbon may be tied up in grains rather than

existing only in the gas phase. Variations in the environmental conditions can also

affect the CO-to-H2 relation. If the metallicity is low, for example, the CO lines will be

weaker for a given H2 density. A high UV radiation field or cosmic ray flux will also be

important because the two molecules can be dissociated, but CO tends to be dissociated

more readily than H2.

All things considered, there are many potential pitfalls in this process. Yet when the

molecular mass is determined via the assumption that the cloud is gravitationally

bound (such a cloud is said to be virialized) and compared to measures of its mass via a

line ratio analysis, the conclusion is that the relationship between H2 density and

CO(J ¼ 1 ! 0) line strength is essentially linear. The relationship is therefore

expressed as,

N H2

cm2

¼ X

Zv

TB½COðJ ¼ 1 ! 0ÞK

dv

km s1

ð9:30Þ

where Eq. (9.30) applies only to the CO (J ¼ 1 ! 0) l 2:6 mm line and X is the sought-

after conversion factor. For the Milky Way, the ‘standard’ conversion factor, X, is in the

range,

X ¼ ð2:3 ! 2:8Þ 1020 cm2 ðK km s1Þ1 ð9:31Þ

and values in this range, in the absence of other information, are widely applied. It is

well-known that X increases if the metallicity decreases, the variation in X being about

a factor of 5 for a metallicity variation of a factor of 10 (Ref. [183]). When galaxies of

different morphological type are taken into account, X may span a range from

ð0:6 ! 10Þ 1020 molecules cm2 ðK km s1Þ1(Ref. [24]).

The CO molecule has now been observed in a wide variety of Galactic environments

as well as in other galaxies. The same molecule that, in excess, is an unwelcome

22The wavelength of this line is long enough that it does not need to be observed from a high altitude (seeFigure 2.2).


constituent of smog on the Earth, ‘lights up’ the otherwise invisible molecular clouds in

our own and other galaxies, providing us with a probe of the nature and properties of

these cold clouds. Moreover, since stars are formed within molecular clouds (Sect.

3.3.2), CO and other molecules allow us to trace stellar birthplaces. The molecular gas

distribution, in general, follows the light distribution of young stars in galaxies23, as

Figure 9.8 shows. The CO emission in the spiral galaxy, M 51 (see also Figure 3.8) is

particularly obvious along the spiral arms where shock waves24 have compressed the

molecular gas, enhancing star formation in these regions. Although the details of star

formation are still not well understood, observations of molecular lines such as from

CO promise to unlock these secrets in the future, as telescope resolution and sensitivity

improve.

Problems

9.1 Compute the frequencies of the following hydrogen transitions and indicate in which

part of the electromagnetic spectrum the resulting spectral lines would be found, (a) Ly d,

(b) H9, (c) BE, (d) H110 a, (e) H239g.

23There are differences on small scales when the distribution is scrutinized in detail, however.24See Footnote 16 in Chapter 3.

Figure 9.8. The galaxy, NGC 5194 (alias M 51 or the Whirlpool Galaxy, see also Figure 3.8) isshown in both these images, from Ref. [133]. At left, is a K-band (Table 1.1) image in greyscalesuperimposed with contours of CO (J ¼ 1 ! 0) emission. The right image shows CO emission onlyusing the same contours as at left. The data were taken as part of the Berkeley–Illinois–MarylandAssociation Survey of Nearby Galaxies (BIMA-SONG). For the both images, the first contour is at 5 Jybeam1 km s1 and, for the right image, each higher contour is 1:585 times higher than theprevious one. The beam has major and minor axes of 5:800 5:100, respectively. The bar on the rightimage is 1 kpc in length. (Reproduced by permission of M. Regan)


9.2 (a) Beginning with DðvÞ (Eq. 9.1), derive the line shape function, DðvÞ, where v is

the velocity of the particles in a gas cloud. Express the result as a function of the velocity

parameter, b, assume that the cloud, itself, is at rest, and that all v c.

(b) From the result of part (a), find an expression for the line width (FWHM) and confirm

that it is the same as Eq. (9.6).

(c) Verify thatR11 DðvÞ dv ¼ 1.

9.3 (a) Find the ratio of natural to thermal line width, nL= nD, for the Balmer H aemission line from a pure hydrogen cloud and typical ISM conditions.

(b) A sodium line (l 589 nm) in the Solar atmosphere at a temperature of 7500 K is

observed to have a line width of 10 nm. Find the thermal to collisional line width ratio,

lD= lP for this line.

9.4 The 44Ti line measured from the Cas A supernova remnant (Figure 1.2) is centred at

1152 15 keV and has a FWHM of 84:6 10 keV (Ref. [78]). What is the velocity of

expansion of this supernova remnant, as measured by the 44Ti ejecta? Does this result agree

or disagree with the maximum known expansion velocity of 10 000 km s1 for this SNR

measured from higher resolution observations?

9.5 (a) With the help of a spreadsheet or other computer software, compute the

frequencies of all hydrogen radio recombination lines (RRLs), Hna, where

50 n 300. Compute the thermal line widths of each of these lines, nD, assuming

that they originate in a typical H II region.

(b) The Very Large Array (VLA) radio telescope25 can be set to any frequency in ‘L’

band over a range 1240 n 1700 MHz. How many Hna RRLs fall within L band?

(c) Once the telescope’s receiver has been set to a given frequency, n0, within L band,

observations at any one time can only be carried out within a sub-band which has some

bandwidth, B, centred at n0, consisting of Nch individual channels of width, nch.

Bandwidth possibilities include – Sub-band 1: B ¼ 50 MHz, Nch ¼ 8, nch ¼ 6250

kHz; Sub-band 2:B ¼ 25 MHz, Nch ¼ 16, nch ¼ 1562:5 kHz; Sub-band 3:B ¼ 6:25

MHz, Nch ¼ 64, nch ¼ 97:656 kHz; Sub-band 4: B ¼ 1:5625 MHz, Nch ¼ 256,

nch ¼ 6:104 kHz. Given these options, what is the highest number of Hna RRLs that

can be observed at any one time and still be spectrally resolved (recall Sect. 2.5)?

9.6 From the information in Figure 9.6 about the ionized gas in the G24.8þ0.1 star

forming region, do the following:

(a) Estimate nðkHzÞ from the line profile.

(b) Estimate the electron temperature, Te, adopting the assumptions given in Sect. 9.4.1.

Correct this value for non-LTE and other effects by increasing it by a factor of 1:5.

25The VLA is operated by the National Radio Astronomy Observatory. The National Radio Observatory is afacility of the National Science Foundation operated under cooperative agreement by Associated Universities,Inc. (USA). Quoted values are 2006 numbers.


(c) Examine and compare Eq. (8.13) with Eq. (9.11) and suggest reasons as to why it

might better to use the continuum measurement to calculate EM, rather than the line

measurement (assuming that the continuum is optically thin).

(d) From the continuum flux density, the corrected electron temperature from (b) above,

the adopted spherical geometry, and assuming uniform density, determine EM, ne, and

MH II.

(e) Assuming Solar abundance, determine the total mass of the ionized gas.

9.7 (a) Plot a graph of the optical line ratios, RO, and RN, as a function of electron

temperature over the range, 5 000 Te 20 000, for two values of electron density,

ne ¼ 102 and ne ¼ 104 cm3. Use a linear scale for Te and a logarithmic scale for the ratio.

(b) Comment on the ability of these line ratios to distinguish between electron tem-

peratures and electron densities over the ranges plotted.

(c) Indicate what the minimum dynamic range (Sect. 2.5) would have to be in order to

measure the ratio, RO, of an H II region with a temperature, Te ¼ 5 000 K.

9.8 Use the information in the question of Example 9.4 to find the H I mass of NGC 2903

using Eq. (9.29) instead of Eq. (9.28). For simplicity, assume that s is about the same at

every velocity in the line.

9.9 From the information of Figure 9.7, find the H I mass (M) of M 81.

9.10 A spherical Galactic H I cloud at a distance of D ¼ 1 kpc has an angular diameter of

y ¼ 1:2 arcmin. This cloud is known to have a l 21 cm line width that is due to thermal

motions only, at a temperature of T ¼ 85 K. The flux density of the cloud at the peak of the

line is fmax ¼ 0:1 Jy. For this cloud, find the following:

(a) The line FWHM, v (km s1).

(b) Its H I mass, MHI (M).

(c) Its diameter, d (pc), and volume, V (cm3).

(d) Its number density, n (cm3).

(e) Its brightness temperature, TB (K), at the line peak.

(f) Its optical depth, t, at the line peak.

9.11 Assuming that the isolated contour at the location, RA = 13h 30m 07:5s, DEC =

471301600 in Figure 9.8, represents a real molecular cloud in the galaxy, estimate the

molecular gas mass (M) of this cloud. Assume that the Rayleigh–Jeans relation holds.


PART VThe Signal Decoded

As the previous chapters have shown, considerable effort is required to extract the basic

parameters of astronomical objects. The signal must be defined, measured, corrected

for instrumental and atmospheric effects, possibly corrected for the effects of inter-

stellar particles and fields, and understood at a physical level that is deep enough to

relate the observed emission to the properties of the source. Now that the basics are all

in place, how dowe actually separate one type of process from the other, and how dowe

piece together reasonable scenarios so that some scientific insight may be gleaned,

beyond a simple description of the astronomical source? The next chapter attempts to

address these questions.

10Forensic Astronomy

. . .a list of observations, a catalogue of facts, however precise, does not constitute science. The

reason is that they tell no story. Only facts arranged as narrative qualify as science.

–Nobel Laureate, John Polanyi (Ref. [126])

In this chapter, our goal is to put together the bits and pieces of information that we

have gleaned from earlier chapters and, as the prologue suggests, discover a story. The

signal may come to us ‘coded’ but, with knowledge of the physics of emission

processes and interactions that occur en route to our telescopes, we may be able to

decipher these codes and gain insight into the nature of the astronomical objects that

have both puzzled and fascinated us. Our tools are not just the sophisticated instru-

ments that detect the signal, but also the equations that relate the light that we observe

to parameters of the source. It is sometimes possible to go even further. By considering

as much evidence as is available, the origin, evolution, energy source, lifetime, relation

to other sources, or other information that goes beyond basic properties, might be

deduced. But first, we need to disentangle the mixed messages that are sometimes sent

to us from space.

10.1 Complex spectra

10.1.1 Isolating the signal

Sometimes we are lucky enough that a signal comes to us uncorrupted and also

unmixed with other types of signals. An example is the 21 cm line from an isolated

interstellar HI cloud. Not only does the radio emission travel through interstellar space

unperturbed by dust, but no other significant emission at or near that wavelength is

emitted by the cloud. It is then straightforward to interpret the result. Often, however,


this is not the case. Even if the signal has not been altered in its long journey towards

Earth, or even if all corrections have been applied so that the signal is as close as

possible to the state in which it was emitted, the result may still include a mixture of

different kinds of radiation.

Consider, for example, the spectrum of NGC 2903 that was shown in Figure 2.3. The

total flux density (f ¼ I, Eq. 1.13) of the continuum emission from this spiral

galaxy is plotted as a function of frequency in this figure. At the lowest frequencies up

to ¼ 1010 Hz, the radio continuum emission declines as a power law with increasing

frequency. From our knowledge of emission processes at radio wavelengths, this

suggests that the radio spectrum is dominated by synchrotron emission due to cosmic

ray electrons in this galaxy. As the frequency increases, we see a broad peak resembling

a Planck curve and, since this peak is in the infrared, this suggests that the emission is

dominated by the collective contributions of dust grains (Prob. 10.1). Above

¼ 1014 Hz, another broad peak is seen, but is less well delineated, given the number

and spread of data points in this part of the plot. This peak extends through the optical

part of the spectrum and we can infer that it is dominated by the collective Planck

curves from different types of stars. Finally, several low intensity points between 1017

and 1018 Hz (the X-ray band) indicate that high energy and/or temperature emission is

also present, likely from a variety of sources.

In fact, the NGC 2903 spectrum is quite typical of what is seen for any normal spiral

galaxy, including the Milky Way. From our knowledge of continuum processes, it is

possible to provide a rather rough interpretation of the global spectrum, as described

above. However, there are many objects in a galaxy, each emitting and thereby

providing information that we would like to obtain. The emission from individual

objects is hidden in plots like Figure 2.3, being swamped by the collective emission of

many objects1. Spectral line emission has also been left out of the figure although the

lines might have been plotted, had the data been available. If a dominant component

can be identified, such as the infrared peak from dust, we are still left with many

questions. What range of grain sizes and temperatures are represented? Is most of the

dust associated with star-forming regions, with old stars, or with molecular clouds?

Answering such questions helps us to grapple with deeper issues, such as the origin of

the dust. We therefore need to isolate the dust (or other) emission from the more

complex global signal.

An obvious approach is to isolate the data spectrally. Since observations in different

frequency bands require different observational techniques, isolating the signal spec-

trally is in fact a byproduct of any observation. This immediately limits the number of

emission processes that are important in the band. Black body radiation from stars is

negligible at centimetre wavelengths, for example. Further narrowing the band may

also help. For example, observations of soft X-rays are more likely to detect emission

from thermal hot gas (Sect. 8.2.3) whereas hard X-rays are more likely to pick up

individual energetic sources, such as X-ray emitting pulsars. Since each radiation

1An exception might be if one type of object clearly dominates the spectrum, such as a very strong active galacticnucleus at the core of a distant galaxy, or a supernova that has recently gone off.

320 CH10 FORENSIC ASTRONOMY

process has its own characteristic spectrum, it is possible to ‘tune’ the observations to

focus on a specific type or types of emission. In the narrow band limit, one tunes to a

specific spectral line.

The data can also be isolated spatially. High spatial resolution allows the targetting

of specific regions or objects for careful study. In external galaxies that are not too

distant, individual HII regions, globular clusters, bright stars, and interstellar clouds

have all been spatially isolated and studied. Although it may not be possible to

isolate individual objects in more distant galaxies, as long as the galaxy itself is

resolved, one may study a region within it whose size scale is that of the resolution

used for the observations. An example has been shown in Figure 2.20, in which a

data cube for the 21 cm emission from NGC 2903 is presented. This line has been

isolated both spectrally and spatially, although individual HI clouds have not been

resolved.

Even with much effort devoted to spectral and spatial resolution, however, it is still

possible for a signal to contain a mixture of different types of emission. It is not always

possible, for example, to arbitrarily improve the spatial resolution of a telescope (see

Chapter 2) and, even if a specific object can be isolated, there may be foreground or

background emission along the line of sight. Different types of radiation may also be

emitted from the same object or spatial region. We must then consider whether there

are other ways to isolate a given signal.

A possibility presents itself if the signals are so different that they can easily be

recognised. An example is when spectral lines sit on top of continuum emission,

such as the radio recombination lines discussed in Sect. 9.4.1 or the X-ray emission

lines seen in a Solar flare (Figure 6.7). The continuum and line can then be dealt

with separately, both providing useful information about the source. Another is if

the signal consists of spectral lines that are Doppler-shifted with respect to each

other (Sect. 7.2.1.2). Various clouds along a given line of sight can then be

separated because their relative velocities prevent them from occurring at the

same frequency. HI and molecular interstellar clouds in our own Galaxy can often

be isolated this way. A more dramatic example is the separation of intergalactic

clouds in the Universe due to the expansion of space, as shown by the Lyman forest in Figure 5.6.

In spite of every best effort, the fact remains that some signals simply cannot be

isolated from others, either because the resolution is insufficient, because there are many

signals from a line of sight at the same frequency, or because several types of signals are

mixed at the source. For such cases – and there are many – we must build a model.

10.1.2 Modelling the signal

The process of building a model spectrum involves guessing at a set of physical

parameters for the source, including any constraints that might already be known,

calculating a model spectrum using these parameters using equations such as those

given in Chapters 8 and 9, and then comparing the model spectrum to the real one.

10.1 COMPLEX SPECTRA 321

Input parameters might include temperature, density, source size, fraction of emission

due to various processes, or whatever else is required to calculate a spectrum. If the

model and real spectra differ from each other, the inputs to the model must be adjusted

and the process repeated until the model spectrum matches the real one as closely as

possible. If a good match can be found, it is then presumed that the input parameters

provide a good description of the source. The problem is often considered to be solved

at this point, although concerns about uniqueness should still be kept in mind. That is, it

is possible that more than one set of input parameters could reproduce the observed

spectrum equally well. For a thorough analysis, the set of input parameters will be

carefully and sequentially adjusted (this is called ‘searching through parameter-space’)

to see if other combinations might also reproduce the spectrum.

An example of a model was shown in Figure 8.9 for the optical continuum emission

from an HII region. This model included contributions from free–bound radiation and

two-photon emission, as labelled in the figure. The difference between the calculated

spectrum and the observed spectrum was then attributed to scattered light from dust.

A more complex example of modelling is illustrated by Figure 10.1 which shows

the total -ray emission from the inner region of our Galaxy obtained from a variety

of -ray instruments (upper data points in the plot). At -ray energies, a wider variety

of astrophysical processes than have so far been described in this text can occur.

From about 10 to 105 MeV, the data are well fitted by a model (solid curve) that

describes diffuse (as opposed to point source) emission, including contributions from

(a) -rays produced by neutral pion ð0Þ decay2, where the pions result from

collisions between high energy cosmic rays (mostly CR protons) and atomic nuclei

of the ISM (mostly protons), (b) Inverse Compton emission (Sect. 8.6) from the

scattering of relativistic CR electrons off of stellar photons, (c) Bremsstrahlung

emission from relativistic CR electrons scattering from interstellar gas (primarily

hydrogen)3, and (d) a small contribution from extragalactic background (EB)

sources (Refs. [164], [165]). In addition, a narrow peak at 0.511 MeV is due to

electron–positron annihilation. However, the emission at energies less than 10 MeV

are not well fitted by this model. At the lower energies, there are likely contributions

to the -ray flux from individual point sources which were not included in the model

of the diffuse emission. A possibility (Ref. [164]) is anomalous X-ray pulsars

(AXPs) – pulsars with extremely powerful magnetic fields. One of the problems

plaguing -ray instruments has been their low spatial resolution (Sect. 2.2.1) which

makes it difficult to pinpoint individual sources. This problem will become less

important as instrumental resolution improves.

2A neutral pion is a subnuclear particle. It is a type of meson which is a particle that consists of a quark–antiquarkpair. Quarks (of which there are different types) are the basic ‘building blocks’ of matter; for example, protonsand neutrons are each composed of three quarks, but the mixture of the types of quarks are different for the twoparticles. The neutral pion is just one type of particle that can result from a high energy collision. It has a rest massof 135 MeV and its most likely decay, after a mean lifetime of only 8 1017 s, is into two -rays.3This is called relativistic Bremsstrahlung and the scattering can be off of protons or other nuclei that still havebound electrons. The spectrum of relativistic Bremsstrahlung emission is not like the thermal Bremsstahlungspectrum discussed in Sect. 8.2 due to the relativistic nature of the interaction and because CR electrons do notfollow a Maxwell–Boltzmann velocity distribution.


To illustrate how the modelling process works, we need to choose a spectral band in

which there are fewer emission processes at work than we have just seen in the -ray

band of Figure 10.1. A good choice is the radio spectrum, for which there are only two

main contributors to the continuum in normal galaxies: thermal Bremsstrahlung

emission and non-thermal synchrotron emission4. Figure 10.2 presents a calculated

radio spectrum, typical of a normal spiral galaxy, which shows the total emission as

well as both the thermal and non-thermal components that contribute to the total.

Throughout most of the plot, the non-thermal component dominates but, at high

frequencies, the thermal component dominates. While this radio spectrum is typical

for a normal galaxy (see Figure 2.3), the specific fraction of emission contributed by

energy, MeV

–310 –210 –110 1 10 210 310 410 510 610

MeV

–1 s

–1 s

r–2

2 E. i

nte

nsi

ty, c

m

–410

–310

–210

–110

1

ICbremss

oπ

total

EB

IC

Figure 10.1. The -ray spectrum of the inner Galaxy (Ref. [164]) showing data (upper points)from a variety of -ray instruments. The ‘intensity’ (number of photons per unit time per unit areaper unit solid angle per unit interval of energy) has been multiplied by E2 on the ordinate axis. Thesolid curve shows the total modelled spectrum which fits the high energy part of the observedspectrum and includes contributions from Inverse Compton (IC) emission, neutral pion ð0Þ decayemission, relativistic Bremsstrahlung (bremss) emission, and the extragalactic background (EB),each of which is labelled. The low energy points are not well fitted by this model (see text). Thedata points at energies below about 2 102 MeV (leftmost on diagram) are somewhat high, likelybecause they were obtained over a slightly different Galactic longitude range. (Reproduced bypermission of Strong, A.W., et al., 2005, A&A, 444, 495)

4However, a dust continuum may be present at the highest radio frequencies.

10.1 COMPLEX SPECTRA 323

thermal and non-thermal components can vary between galaxies and the value of NT,

which is related to the cosmic ray energy spectrum (Eq. 8.43), can also differ between

galaxies. How, then, can we determine the thermal/non-thermal contributions and

extract NT from the observations? Example 10.1 suggests an approach.

Example 10.1

Assume that the total continuum emission from a galaxy at radio wavelengths consists only ofthermal Bremsstrahlung radiation whose spectrum can be described by IðTÞ ¼ a 0:1

(Eq. 8.14) and non-thermal synchrotron radiation whose spectrum is IðNTÞ ¼ b NT

(Eq. 8.42), both optically thin. Describe a way to determine the constants, a, b, and NT ,when only the total emission is measured.

.1e–2

.1e–1

.1

1.

.1e2

1e+07 1e+08 1e+09 1e+10 1e+11 1e+12

Frequency (Hz)

Ι ν

Ι –0.1ν ∝ ν

Ι α NT ∝ νν

Ιν (total)

Ι ν ∝ ν

Ιν (total)

Figure 10.2. A model radio spectrum, showing the total observed emission ðIðtotalÞ / Þ, andits two contributors, a thermal Bremsstrahlung component ðI / 0:1Þ and a non-thermalsynchrotron component ðI / NT Þ. The observed spectral index, , varies with frequency.


The total emission can be represented by the sum of the thermal and non-thermal

components (see also Figure 10.2),

IðtotalÞ ¼ IðTÞ þ IðNTÞ ¼ a 0:1 þ b NT ð10:1Þ

There are three unknowns, a, b, and NT, and therefore three equations are required. These

equations can be set up by measuring IiðtotalÞ at three different frequencies, i, and writing

Eq. (10.1) for each frequency. The set of three equations can then be solved for a, b, and

NT. Note that a similar equation could be written for the flux density (Prob. 10.2) since

f ¼ I (Eq. 1.13) and we take to be constant.

The constant, a, is related to the physical quantities associated with thermal Bremsstrah-

lung emission via Eqs. (8.13) and (8.14) so, depending on what other information is

available, it may then be possible to extract information about the thermal part of the source.

Similarly, b and NT are related to the non-thermal parameters by Eqs. (8.36) and (8.42).

With the help of computer software, this concept of modelling can be extended to

‘build’ spectra of nebulae, galaxies, or whatever other system is of interest. For

example, there are many stars within any given resolution element in a typical optical

observation of a galaxy. Extracting knowledge of the admixture of different kinds of

stars requires modelling the total spectrum by including different numbers of stars of

various spectral types (e.g. Prob. 10.3) until a good match results. This approach is

called population synthesis and provides important information about the galaxy. For

example, if the spectrum can be modelled without including any hot massive stars, then

the galaxy is not undergoing any significant star formation at the present time (Sect.

3.3.4). On the other hand, if many hot massive stars are required to describe the

spectrum, then the galaxy is an active star-forming system. Extending this modelling

concept to include HII regions, dust, or other contributors to the spectrum is also

possible and is currently actively pursued by researchers interested in knowing the

make up of galaxies.

As a final comment, we note that the process of model-building is not restricted to

spectra. The spatial distribution of light can also be modelled, something that is

routinely carried out over a variety of scales, from planetary systems, to large scale

structure in the Universe. Modelling spectra, however, is a particularly productive

approach, since there are only a limited number of known emission mechanisms and

corresponding spectral behaviour.

10.2 Case studies – the active, the young, and the old

How can we build links between objects and events in astrophysics and what scenarios

can be pieced together? A good approach is to focus on specific objects or regions and

apply the astrophysical principles that we have already seen. To this end, we now

present case studies of three very different objects.

10.2 CASE STUDIES – THE ACTIVE, THE YOUNG, AND THE OLD 325

10.2.1 Case study 1: the Galactic Centre

Eight kpc from the Sun in the direction of the constellation, Sagittarius (the Archer) is

the location of the Galactic Centre (GC, see also Figure 3.3). At optical wavelengths,

the GC is obscured by many magnitudes of extinction from dust along the Galactic

plane. However, at infrared, radio and hard X-ray wavelengths, which are not grossly

affected by dust, it is possible to catch a glimpse of the heart of our Galaxy. ‘Galactic

Centre’ is loosely used to refer to the very nucleus of the Milky Way as well as the

vicinity around it, of order a few hundred pc, within which the properties differ

markedly from the rest of the Galaxy. Figure 10.3 shows two views at radio wave-

lengths. These images show a rich field containing many sources. Some sources are

unique to the GC, such as the magnetically aligned threads and the outflow-related shaped Galactic Centre radio lobe (GCL). Both non-thermal (synchrotron) and thermal

(Bremsstrahlung) radio emission are present.

The GC harbours a richer variety of sources and processes than even Figure 10.3

suggests. In this region, there is hard X-ray emission, X-ray and IR flaring, strong

organised magnetic fields, high stellar densities, high gas pressures and temperatures,

Figure 10.3. The Galactic Centre at two different radio frequencies. In each image, the Galacticplane runs from top left to bottom right. On the left is the 90 cm image from the Very Large Arrayradio telescope in New Mexico, USA (resolution, 20, Ref. [25]). There are many supernova remnants inthis image, as well as narrow threads of emission, likely due to aligned magnetic fields. On the right isthe 3 cm image from the Nobeyama telescope in Japan (resolution, 30, Ref. [72]). The Galactic Centreradio lobe, with a diameter of about 200 pc, forms an shape perpendicular to the Galactic planedelineated here with a dashed curve. This feature is brighter at 3 cm than at 90 cm and thereforehas a flat spectral index. (Reproduced by permission of C. Brogan)


and high velocity dispersions. Given that the density of the Galaxy, as a whole,

increases towards the centre, it is not surprising that the properties of the GC are

more extreme than elsewhere in the Galaxy. In addition, however, there is also an

‘activity level’, as attested to by the presence of very high energy radiation and

features, like the GCL, that are suggestive of outflows. Other outflow features, seen

over a variety of scales, include bipolar X-ray lobes centred on the nucleus, loops of

molecular gas (Ref. [61]), and an extended column of -ray emission above the nucleus

from electron–positron annihilation referred to as the annihilation fountain. Although

the origin of some features (such as the annihilation fountain) is still debated, put

together, the collective observations imply that the GC harbours an active galactic

nucleus (AGN), similar to those observed in other galaxies and discussed at various

times throughout this text (e.g. see Figs. 5.5 and 8.16). The AGN is located at the

nucleus of the Milky Way – a source called Sgr A5. This is the closest AGN to us and

its proximity allows us to study it with a linear resolution that surpasses anything

possible for external galaxies.

The AGN paradigm requires the presence of a ‘driver’ for the high energies that are

associated with it. Sometimes referred to as a central engine, sometimes a monster, the

object that is believed to be powering the AGN is a supermassive black hole. Although

the black hole itself does not emit light, the strong gravitational field around it (see

Sect. 7.1) provides the energies required to drive the other observed phenomena (e.g.

Sect. 5.4.2). For the Milky Way AGN, a black hole mass of a few million solar masses

is implied. What is the evidence for the existence of this black hole? Is it an argument

based only on the need for a high energy source or are there any other lines of evidence?

Example 10.2 outlines the case.

Example 10.2

Figure 10.4 (Ref. [150]) shows, on the left hand side, a 2 mm (near-IR) view of a star field

in the GC that is only 2 arcsec on a side. For a distance to the GC of D ¼ 8 kpc, the

corresponding linear scale is d ¼ 0.078 pc (Eq. B.1). Almost all of the stars in this image

belong to a dense stellar cluster that is near the Galactic core, since the probability of

finding a disk star (one that is closer to us along the line of sight) in such a small field of

view is small (Prob. 10.4). A star, S2, is labelled as well as the position of Sgr A.

The proper motion of S2 has now been measured and, in fact, traced over a 10 year period

so that its orbit about Sgr A has been precisely determined, as shown in the right hand

image. The velocity of S2 is highest at closest approach to Sgr A6, indicating that the

interior mass is concentrated at this position. The true orbit, which has some inclination

with respect to the plane of the sky, is an ellipse (Appendix A.5) with Sgr A at one focus.

The apparent (observed) orbit is also an ellipse, but Sgr A is no longer at the focus of the

apparent orbit because of the projection onto the plane of the sky. This fact can be used,

with some geometry, to determine the true orbit. For S2, it is found that the inclination

5Surprisingly Sgr A* itself is actually underluminous for an AGN of its mass.6Recall Kepler’s laws from Sect. 7.2.1.2.


is i ¼ 46, the eccentricity is e ¼ 0:87, the semi-major axis is a ¼ 5:5 light days

ð1:42 1016 cmÞ, and the orbital period is T ¼ 15:2 yr.

The values of a and T determined from the orbit, together with Eq. (7.5), yield a mass

interior to the orbit of, M ¼ ð3:7 1:5Þ 106 M. The separation of S2 from the nucleus

at closest approach is only 17 light-hours (124 AU). The only known object that could

contain this much mass in such a small volume is a supermassive black hole. A stellar

cluster or any other conceivable types of object would require a much larger volume.

Since the time of this definitive observation, additional constraints have been placed on

the black hole. From Doppler shifts of spectral absorption lines in the orbiting star, the

star’s radial velocity, vr, has been determined and, together with the measured transverse

motion on the sky, vt, this has allowed a determination of its true space velocity (see

Figure 7.2). In addition, orbits for more stars have now been determined. The result has

been a more accurate and precise black hole mass of Mbh ¼ ð3:7 0:2Þ 106 M(Ref. [63])7. In addition, an X-ray flare has been observed from Sgr A which showed

some variability on a timescale of only 200 s (Ref. [127]). This places a limit on the size of

the black hole of d < 0:4 AU, via Eq. (7.21). Thus, 4 million Solar masses are within a

region that is smaller than the orbit of Mercury! The conclusion that the Milky Way

harbours a supermassive black hole at its centre seems inescapable.

Figure 10.4. Left: the star field around the nucleus of our Galaxy, shown in this 2:1 mm (near-IR)image over a field of view, 2 arcsec on a side with an angular resolution of 0.06 arcsec. The nucleus,located at the position of Sgr A and a star called S2, whose light is blended with Sgr A, are bothmarked. Right: The elliptical orbit of S2 about Sgr A (as seen on the sky) is shown, after havingbeen observed over the course of 10 yr. Notice that the speed of the orbit is greatest at closestapproach. (Reproduced by permission of ESO)

7The result is distance dependent. For a GC distance of 8.5 kpc (Table G.3), Mbh ¼ ð4:4 0:2Þ 106 M.


10.2.2 Case study 2: the Cygnus star forming complex

In the constellation of Cygnus (the Swan), within the disk of the Milky Way and about

1.8 kpc distant, is the Cygnus star forming complex. A picture of this region, from

combined radio and far-infrared images, is shown in Figure 10.5. The enormous

angular size of this region is truly impressive. The diameter of the field of view covers

fully 10 of sky, about 20 times the diameter of the full moon. If our eyes were sensitive

to the wavebands shown, it is clear that the sky would look nothing like the star filled

firmament that we see each night.

The Cygnus star forming complex is a vast stellar nursery in which stars are born

from dense molecular gas, live their lives, and die, recyling their chemically enriched

material back into the interstellar medium. In the process, winds from hot massive stars

(e.g. Sect. 5.4.2), radiation, and explosive energy from supernovae compress, heat,

ionize, and disrupt the surrounding gas clouds. For a star-forming region in the disk of

the Milky Way, this is one of the most active. How can such a picture about this region

emerge? For this, we need to draw on a number of resources, including the information

contained in Figure 10.5, data from other wavebands, and a knowledge of stellar

Figure 10.5. A complex star forming region in our Milky Way in the direction of the constellation,Cygnus. The image diameter spans 10 of sky. In this image, the colour rose represents emission at74 cm, green is 21 cm emission, turquoise is 60 mm emission, and blue is 25 mm emission, allcontinuum emission. Two supernova remnants are labelled. [Credit: Composed for the CanadianGalactic Plane Survey by Jayanne English (CGPS/U. Manitoba) with the support of A. R. Taylor(CGPS/U. Calgary)] (see color plate)


evolution that has been painstakingly pieced together from both observation and theory

(see Sect. 3.3). We begin with a discussion of Figure 10.5 given in Example 10.3.

Example 10.3

In Figure 10.5, four different wavelengths of continuum emission are represented in false

colour: the radio wavelengths, 74 cm ð ¼ 408 MHzÞ in rose/red and 21 cm

ð ¼1420 MHzÞ in green, and the far-infrared wavelengths, 60 mm in turquoise and

25mm in blue. Careful quantitative analysis of the individual images that contribute to

Figure 10.5 is needed to properly extract the information that it contains. Nevertheless, it is

possible to piece together some information, with just a careful visual examination. The

mix of colours of the various sources, for example, provides us with clues about the

emission processes, and therefore about the most likely sources.

Any object that is reddish or yellow-red, depending on other emission in the vicinity,

must be strongest at the longest radio wavelength, 74 cm. Since optically thin synchrotron

emission becomes stronger at longer wavelengths, this suggests that the reddish sources are

steep-spectrum, optically thin synchrotron emitting sources. Two such circular features are

labelled in the figure. The source, SNR Cygni, so named because it lies near the star,

Cygni, spans 1 of sky. As the name indicates, this is a supernova remnant, which we

could infer from its non-thermal radio spectrum and bubble-like shape. A second labelled

SNR, G84.2-0.8, is also quite distinct and shows the edge-brightenedmorphology expected

from a shell of material.

Throughout the image are also many reddish (in false colour) point sources. These may

look like stars, but they are clearly not, since their reddish colour indicates that they emit

most strongly at the longest radio wavelength, again consistent with a synchrotron

spectrum. Stellar continuum spectra, by contrast, are Planck curves that peak from near-

IR to UV wavelengths (Tables G.7, G.8, and G.9, and Eq. 4.8) and stellar emission would

be negligible at all wavelengths shown in the figure. The red point sources are more broadly

distributed in the image (more obvious in a wider field of view), rather than clustered in the

Cygnus star-forming complex, and they are unresolved. This information suggests that

these are background sources, likely quasars, in the distant Universe.

As the emission shifts to green ð 21 cmÞ and blue, more blending is seen between

colours and sources. The green emission emphasizes the contribution from flat-spectrum

thermal Bremsstrahlung emission (see Figure 10.2), implying the presence of ionized gas

(HII). In fact, ionized gas is abundant in this complex, both as diffuse emission throughout

the region as well as discrete cocoon-like ‘knots’ in the map. The existence of HII regions

requires the presence of hot O and B stars, as described in Sect. 5.2.2, and also means that

there is sufficient interstellar gas available to be ionized.

Finally, the emission from the far-infrared wavelengths, shown in turquoise and blue,

permeates the region. At these wavelengths, we are seeing collective Planck curves from

dust (e.g. Sect. 10.1.1). At 60 mm and 25 mm, the characteristic temperatures (Eq. 4.8)

are T ¼ 48 K and T ¼ 116 K, respectively, both far too cool to represent photospheric

emission from stars. Some very blue ‘knots’ of emission are also seen, possibly very small

dusty regions around forming stars.


Thus, in this image alone, we see supernova remnants, abundant interstellar gas and dust,

and HII regions. We can also infer the presence of hot, O and B stars, although they are not

seen directly. From our knowledge of stellar evolution, we know that HII regions, O and B

stars and supernovae are all tracers of recent star formation (Sect. 3.3.4) since the lifetimes

of O and B stars are so short (of order 106 yr). They should also, therefore, still be

associated with the molecular material from which they formed. Moreover, since dust

and molecular gas are correlated spatially in the Milky Way (e.g. see Figure 3.20), this also

suggests that molecular gas should be present throughout this region.

The rudimentary treatment of Example 10.3 should never take the place of a proper

quantitative analysis. Yet, with some careful scrutiny and an understanding as to which

emission processes dominate in various wavebands, it is sometimes possible to extract

useful information. Are there any other data, then, that support the conclusions of

Example 10.3?

The Cygni SNR has been detected at X-ray and -ray wavelengths (Ref. [173]),

confirming the high energy nature of this source. Its expansion velocity has also been

measured to be vexp ¼ 900 km s1 and the result of the expansion analysis indicates

that the supernova exploded some 5000 yr ago. This is a very recent event, by

astronomical standards.

The molecular gas, inferred to be present, has been directly observed and is extensive

throughout the region in a giant molecular cloud (GMC) complex. From CO observa-

tions (see Sect. 9.5.1), the estimated total molecular gas mass is MGMC ¼ 5 106 M(Ref. [148]), making it one of the most massive molecular cloud complexes known in

our Galaxy.

Finally, the hot O and B stars themselves, assumed to be responsible for the thermal

radio emission, have also been directly observed. Several star clusters containing O and

B stars, called OB associations are embedded in the cloud. One of them, the Cygnus

OB2 association, is one of the most massive and richest clusters in the Galaxy,

containing 2600 cluster member stars of which about 100 are O or B stars (Ref.

[86]). The total mass of this cluster alone is is 410 104 M.

There is little doubt that this complex plays host to a rich array of processes, of which

star formation and the evolution of massive stars are key elements. While the details of

star formation are not completely understood, complexes like the Cygnus cloud

provide astrophysical ‘laboratories’ that may reveal the secrets of star formation in

the future.

10.2.3 Case study 3: the globular cluster, NGC 6397

In the southern hemisphere constellation of Ara (the Altar), located out of the disk of

our Galaxy is NGC 6397, one of the closest globular clusters to us ðD ¼ 2:6 kpcÞ. The

globular clusters are spherical in appearance, typically contain a few hundred thousand

stars each and have a typical absolute magnitude of MV ¼ 7:3. There are about 100


known that dot the halo region and bulge of our Milky Way (see Figure 3.3). These

clusters are believed to be the oldest stellar systems in the Universe, and NGC 6397

is one of the oldest, at an age of 13.4 Gyr (Ref. [67]). This is only marginally less

than the age of the Universe (13.7 Gyr, Sect. 3.1), and the two values are in

agreement if typical error bars of 0.8 Gyr are taken into account. Thus, these sytems

must have formed in the very early Universe and would have been among the first

types of objects that took part in the assembly of massive galaxies like the Milky

Way. They provide the means of dating the Milky Way, itself, whose age (13.6 Gyr

old) is taken to be the age of the oldest clusters plus some allowance of time for the

clusters to form. What are the characteristics of this sytem that lead to these

conclusions? We begin, in Example 10.4, with a discussion of the colour-magnitude

(CM) diagram of NGC 6397, shown in Figure 10.6.

Example 10.4

It is instructive to compare the CM diagram of the globular cluster, NGC 6397(Figure 10.6), with that of a collection of stars in the gaseous dusty disk of the Milky

Way in the Solar neighbourhood (Figure 1.14). If the mix of stars in the globular cluster

were the same as in the disk, there should a close resemblance between the figures8.

What we find, however, is that the two CM diagrams differ significantly from each

other.

0.8 1.0 1.2 1.4 1.6(v−y) [mag]

19

18

17

16

15

14

13

12

V [m

ag]

NGC 6397

Figure 10.6. Colour-magnitude diagram of the globular cluster, NGC 6397 (Ref. [91]), taken withthe Danish 1.54 m telescope in La Silla, Chile. Crosses mark specific stars that were targetted forspectroscopy in this study. (Reproduced by permission of Korn, A., 2006, astro-ph/0610077, TheMessenger, 125, 2006, 6–10)

8Note that the filter bands of the two figures differ, but this does not change the argument.


First of all, the globular cluster CM diagram lacks any trace of hot, luminous massive

stars which we know, from their short lifetimes, to be young. All stars on the main sequence

occur below a turn-off point which is the location that the main sequence turns off into the

red giant branch (see Sect. 3.3.2). If any recent star formation had been occurring in NGC

6397, then there should be evidence of hot, massive stars in this diagram, and they should

have been be easily detected, being more luminous than the other detected stars on the

lower main sequence9. From a knowledge of stellar evolution, we know how long a star can

remain on the main sequence before it exhausts its fuel and turns off to the red giant branch.

This fact can be used to determine how long it has been since stars formed in the cluster.

Model colour-magnitude diagrams for a range of ages have been constructed from theory

and they are then compared to the data, the best-fit result giving the cluster age. This is

called cluster fitting.

Secondly, the turn-off point is quite well defined and narrow, unlike the broad turn-off

region in Figure 1.14. For NGC 6397, the turn-off corresponds to a stellar temperature of

Teff ¼ 6254 K (Ref. [91]). If the stars that formed long ago had formed continuously over

some period of time, then different stars of different temperatures would now be turning off

the main sequence, thickening the turn-off position. A sharp turn-off point, then, indicates

that all the stars in the cluster were formed in a relatively short period of time. In other

words, the cluster itself has a unique age.

Other evidence, besides the cluster fitting technique described in Example 10.4, also

points to the conclusion that globular clusters are old systems in which star formation

ceased long ago. There is currently no atomic or molecular gas in globular clusters and

therefore no supply of material from which stars could form. Images of NGC 6397 (e.g.

Figure 10.7) reveal only a population of old stars and stellar remnants. White dwarfs,

for example, are the relics of stars that have long since left the main sequence and come

to the end of their nuclear burning lifetimes. A significant population of white dwarfs

has been detected in NGC 6397 to an extremely faint magnitude limit (see Figure 10.7).

Since the luminosity of a white dwarf declines with time as it cools, the location of the

white dwarf cooling curve on the CM diagram also allows the cluster to be dated, with

consistent results.

Other creative dating techniques have also been applied. For example, observations

of the element, beryllium, in cluster members are useful. Beryllium is neither produced

in the Big Bang, nor by nucleosynthesis in stars in any significant quantity. The source

of its production is by spallation from cosmic rays (Sects. 3.3.4, 3.6.1). Over a period of

time, the quantity of beryllium is expected to increase in any gas from which the cluster

originally formed. Therefore a measure of the beryllium abundance in the cluster

provides a way of timing when the cluster formed, with some assumptions related to

calibration. Beryllium absorption lines in the atmospheres of two stars in NGC 6397

9This cluster does, however, contain a population of blue stragglers, which are stars that are unexpectedly blue,given the age of the cluster (Ref. [10]). The most likely explanation for blue stragglers is that they have beenformed by stellar collisions in the dense cores of these clusters.


have now been measured, resulting in an age estimate that is in agreement with the

other values (Ref. [118]).

Another important clue as to the age of a globular cluster is its very low

metallicity (Sect. 3.3.1). For NGC 6397, for example, the metallicity is only

1/100th of the value observed in the Sun. Since we know that successive genera-

tions of star formation will increase the metallicity over time, this implies that

globular clusters formed early and that successive generations of stars have not

occurred within them. As pointed out in Sect. 3.3.4, however, it is important to note

that the metallicity is not zero. This means that some metal enrichment took place

even before the formation of the globular clusters. A population of massive stars

that rapidly exploded as supernovae could have accomplished this, but the exact

nature of this population is unknown10.

10 This population is referred to as ‘Population III stars’.

Figure 10.7. Hubble Space Telescope view of the centre of the globular cluster, NGC 6397,shown with exquisite detail. This image reaches the faintest magnitudes ever observed for lowmass stars and white dwarfs. Two squares mark fields that are blown up at the right. In the imageat the top right, a faint white dwarf star is circled. This white dwarf is at 28th magnitude andis blueish in colour, indicating its hot surface temperature. In the lower right image, a faintred dwarf star, at 26th magnitude, is circled. Close scrutiny will reveal faint background galaxiesthat are seen through the cluster. [Reproduced by permission of NASA, ESA, and H. Richer(University of British Columbia)]


All indicators are that there was an early rapid burst of star formation at which time

the globular clusters formed, and star formation then ceased within these systems.

Current data suggest that the globular cluster formation epoch began within about

1.7 Gyr of the Big Bang, lasted for 2.6 Gyr, and ended 10:8 1:4 Gyr ago (Ref. [67]).

The spatial distributions and velocities of globular clusters in the spherical halo of the

Milky Way agree with this scenario, pointing to an early formation period in low

metallicity gas, before material settled into the spiral disk in which we now do see

active star formation.

It is clear that globular clusters contain information whose implications go far

beyond the boundaries of the cluster, itself. These systems provide clues as to the

conditions that were present in the very early Universe at the time that our Milky

Way was just forming. NGC 6397 has, in addition, proven to be a laboratory for the

study of the lowest mass stars. The faint end of the stellar main sequence in NGC

6397 has been completely detected, showing that the main sequence has a low

luminosity cut-off (see Figure 10.7). From these observations, a low mass cut-off of

0.083 M has been determined (Ref. [135]). This is in beautiful agreement with the

theoretical low mass limit of Mmin¼ 0:08 M required for the nuclear burning

discussed in Sect. 3.3.2. It also hints that there may be a population, yet to be

detected, of sub-stellar brown dwarfs in which nuclear burning has not, and never

will, occur.

10.3 The messenger and the message

In 1610, Galileo Galilei published his Sidereus Nuncius (‘Sidereal Messenger’ or

‘Starry Messenger’) which was the first book to be published based on observations

of the sky through a telescope. In it, he described the fact that the Moon is not

perfectly spherical but has an irregular surface, the fact that Jupiter has four moons

(today called the ‘Galilean satellites’) which orbit about it, and that many neb-

ulosities, such as the apparently fluid light of the Milky Way, consist of a myriad of

stars that are simply unresolved by eye. In this brief report, he upset an Aristotelian

world view that held the heavens to be ‘perfect’ and showed that the Earth was not

at the centre of every orbit. Instead of deferring to past authority, he was willing, to

paraphrase the Prologue to Chapter 8, to ‘see a thing, and say that he sees it’. For

this he suffered and, for this, we have gained, for by so doing, he laid the

foundations of modern science.

The ‘signal’ that we have tracked over the scope of this text has not been imbued with

meaning, though the meaning may indeed be there for those who seek it. What has been

assumed, implicitly, is that it is actually possible to measure a signal – a message – and

discern something concrete about the objects that have emitted it or altered it along the

way. For this, we require theoretical models that, if not perfect, at least provide as close

a description as possible of our physical world.

The achievements and power of this approach are impressive. The spins of some

electrons in a distant galaxy change their orientation, and we learn that the galaxy’s

10.3 THE MESSENGER AND THE MESSAGE 335

mass is one hundred billion times that of the Sun. Electrons alter their course slightly

while passing by nuclei, and we find that the hot gas in a cluster of galaxies far

outweighs the mass of the visible stars. Atoms in the atmospheres of stars show

absorption lines that are shifted slightly in frequency, and we conclude that a super-

massive black hole lurks at the core of the Milky Way. Slight warps in space–time

perturb photons travelling through a cluster of galaxies implying that dark matter

dominates all other material. And glimmers of light from the dawn of time hint of

the origin and evolution of the Universe on vast cosmological scales. Perhaps in the

future, such a world view may be superseded by something even more perfect and

profound. For now, the signals that link our detectors to the cosmos continue to yield

their secrets.

Problems

10.1 The infrared peak in the spectrum of NGC 2903 is due to the collective contributions

of many different types of grains over a variety of temperatures. Figure 2.3, however,

suggests that the bulk of the grains might be described by one ‘characteristic’ temperature.

Estimate this temperature from the figure.

10.2 At the three frequencies, 4850 MHz, 2700 MHz, and 1465 MHz, the radio conti-

nuum flux densities of a galaxy are, respectively, f4850 ¼ 3:94 Jy, f2700 ¼ 5:37 Jy, and

f1465 ¼ 7:69 Jy.

(a) Write three equations in three unknowns that show the explicit contributions of

thermal Bremsstrahlung and non-thermal synchrotron radiation to the total emission (see

Example 10.1, and note that cgs units are not necessary).

(b) With the help of computer algebra software, solve for the three unknown constants.

[Hint: Include the restriction, NT < 0:15.] Specify the units of each of the constants.

(c) What fraction of the total emission is due to non-thermal emission at ¼ 4850 MHz

and at ¼ 1465 MHz?

10.3 With the help of a spreadsheet or other computer software, calculate the shape of the

total spectrum of a small galaxy that consists only of three different types of stars: 106 M5V

stars, 5000 A0V stars, and 20 K0Ib stars (see Tables G.7, G.9). Since we can only measure

the flux density of stars, not their specific intensity (e.g. Figure 1.9), the black body curves

of the stars need to by multiplied by R2, where R is the stellar radius [f / R2BðTÞ, Eqs.

1.10, 1.14] in order to see the shape of the spectrum. Plot each of the three stellar

components as a function of frequency as well as the total. You may wish to change the

admixture of stars to see how the result varies.

10.4 If we take the number density of disk stars in the Milky Way to be approxi-

mately n ¼ 0:1 pc3, how many disk stars (on average) should be visible in the left

image of Figure 10.4?


10.5 Of the distinct features in Figure 10.3, identity one that is likely to be an HII region,

and explain why.

10.6 For the following radio sources, indicate whether the emission is due to synchrotron

radiation, thermal Bremsstrahlung radiation, or whether it is possible to tell. Provide an

explanation.

(a) A source that has a steep, negative power law spectrum.

(b) A source that has a flat spectrum and is polarised.

(c) A source that has a flat spectrum and is not polarised.

10.7 Indicate whether the following lines are more likely to have originated in an HI

cloud or an HII region and provide a reason. (When ‘absorption’ is indicated, assume that

the required background source is available.)

(a) The 21 cm line in absorption.

(b) The 21 cm line in emission.

(c) The Ly line in absorption.

(d) The Ly line in emission.

(e) The H line in absorption.

(f) The H line in emission.

10.8 Suggest a possible object or objects that could produce spectra with the following

characteristics (each part represents a different object or objects):

(a) There are many spectral lines in the mm-wave part of the spectrum. The under-

lying continuum shows a slope that suggests a Planck curve which peaks in the

infrared.

(b) The spectrum contains a broad Planck-like peak in the near-infrared and optical. No

other significant continuum is observed.

(c) The optical emission resembles that of a star and is polarised.

(d) The X-ray continuum shows a declining spectrum with increasing frequency. Some

X-ray emission lines are seen.

(e) The radio spectrum is flat and has emission lines.

(f) An emission line is observed at a wavelength of 21:232 cm.

(g) Numerous optical absorption lines are observed against a Planck continuum.

10.9 Using the concept of minimum energy (see Sect. 8.5.3), ‘hot spots’ in the radio lobes

of Cygnus A have been found to have lifetimes (at ¼ 89 GHz) of t ¼ 3 104 yr. From the

information given in Figure 8.16, determine whether relativistic particles in the hot spots

10.3 THE MESSENGER AND THE MESSAGE 337

could have been accelerated at the nucleus of the galaxy or whether they were accelerated

in the lobe itself.

10.10 Visit the Astronomy Picture of the Day (APOD) website at http://antwrp.gsfc.na-

sa.gov/apod/astropix.html and, as we did in Example 10.3, attempt to identify the type of

emission that is displayed in an image that interests you. Some questions to ask might be:

(a) Does this picture represent a single waveband or is it a composite?

(b) Does the image show continuum or line emission?

(c) Is the emission thermal or non-thermal in nature?

(d) If a spectrum in the displayed waveband were available to you, what information

could you obtain about the source?

10.11 Visit the Multi-wavelength Milky Way website at http://adc.gsfc.nasa.gov/mw/. For

each of the wavebands shown, (a) identify the important emission mechanism(s), and (b)

provide examples of the types of sources that contribute to the observed emission.

10.12 A point source of magnitude, V ¼ 18, is observed in a galaxy at a distance of

D ¼ 30 Mpc. Is this source an individual star, a globular cluster, or a supernova?


Appendix AMathematical and GeometricalRelations

The German mathematician Carl Friedrich Gauss suggested planting avenues of trees in the barren

plains of Siberia to form a giant right-angled triangle. He was attempting to signify to any

watching Martians that there was life on Earth intelligent enough to appreciate the wonders of

geometry.

–The Science of Secrecy, by Simon Singh

A.1 Taylor series

A Taylor series is an expansion about a point, a,

f ðxÞ ¼ f ðaÞ þ f 0ðaÞðx aÞ þ f 00ðaÞ2!

ðx aÞ þ . . .f ðnÞðaÞn!

ðx aÞn þ . . . ðA:1Þ

where f 0ðxÞ dfdx

, f 00ðxÞ d2fdx2, etc. If a ¼ 0, then the expansion is called a Maclaurin

series.

A.2 Binomial expansion

For any value of p, integer or non-integer, positive or negative, one can write,

ðaþ xÞp ¼ ap þ p aðp1Þ xþ p ðp 1Þ2!

aðp2Þ x 2 þ . . .þ xp ðA:2Þ

For example, if x 1, then ð1 þ xÞ1=2 1 þ 12x


A.3 Exponential expansion

e x ¼ 1 þ xþ 1

2!x 2 þ . . .þ 1

n!xn ðA:3Þ

A.4 Convolution

When two functions are convolved together, they are effectively ‘blended’ with each

other. Convolutions arise in a wide variety of situations in which the signal is observed

with an instrument that imposes some limitation upon what can be detected. For

example, we can consider the situation in which a source on the sky is observed

with an optical telescope. The true brightness distribution of this source is arbitrarily

sharp since it is not hindered by the spatial resolution of our instruments. When we

observe it, however, the true brightness distribution is convolved, at every position on

the sky, with the point spread function (PSF, see Sect. 2.3.2) of the telescope, normal-

ized so that the total flux of the source is preserved. The crisp image, such as that shown

in Figure 2.11a, becomes blurred, as suggested in Figure 2.11b, because of this

convolution. Thus, convolution ‘smooths’ the source brightness distribution.

Mathematically, if hðxÞ is the convolution of the two functions, f ðxÞ and gðxÞ, in one

dimension, x, then,

hðxÞ ¼ f ðxÞ gðxÞ Z 1

1f ðuÞ gðx uÞ du ðA:4Þ

the operator, , meaning convolution. Convolutions are commutative, such that,

f ðxÞ gðxÞ ¼ gðxÞ f ðxÞ. The result of the convolution, hðxÞ, is a function of position

along the one-dimensional coordinate (x). The right hand side of Eq. (A.4) indicates

that one of the functions, say the one-dimensional PSF, is shifted across the one-

dimensional source brightness distribution with a multiplication at each shift and then

the result is integrated over all shifts. For a real situation in which there are two

coordinates, x and y, the convolution must be carried out in two-dimensions.

A.5 Properties of the ellipse

It is useful to review the properties of an ellipse since orbital motion follows the path of

an ellipse. In addition, circular disks, such as that of a spiral galaxy or proto-planetary

disk, become ellipses when projected onto the sky. The resolution (beam) of a

diffraction-limited telescope (see Sect. 2.2.4) may also be elliptical on the sky.

An ellipse is defined as that set of points that satisfies the equation, r þ r0 ¼ 2 a,

where a is a constant called the semi-major axis. The distances, r and r0 are distances

measured from a point in the set to the two focal points, F and F0 of the ellipse,

340 APPENDIX A MATHEMATICAL AND GEOMETRICAL RELATIONS

respectively. In the case of orbital motion in which a small mass, m is orbiting a large

mass, M, one focal point, F , will be at the location of M and the other focal point F0

will be empty. The distance of each of the two focal points from the centre of the ellipse

is the quantity, ae, where e is called the eccentricity. If e ¼ 0, then the ellipse reduces to

a circle and the two focal points become coincident at the centre of the circle. If e ¼ 1

then the ellipse ‘opens up’ and becomes a parabola. Thus, 0 e < 1 is a requirement

for an ellipse, the circle being a special case, with highly elongated ellipses having

larger values of e. Table A.1 provides additional properties.

Table A.1. Properties of the ellipse

Description Expression

Area of the ellipse A ¼ p a b

Relation between a and b b2 ¼ a2 ð1 e2ÞDistance of closest approach to F a rmin ¼ a ð1 eÞDistance of farthest approach to F a rmax ¼ a ð1 þ eÞDistance from a point on the curve to F b r ¼ að1 e2Þ=ð1 þ e cos uÞaFor an object orbiting the Sun, rmin is called perihelion and rmax is called aphelion.Similarly, closest approach to a star would be periastron and farthest is apastron, or for agalaxy, perigalacticon and apogalacticon, respectively.b The angle, u, is measured around the ellipse from u ¼ 0 at r ¼ rmin.

A.5 PROPERTIES OF THE ELLIPSE 341

Appendix BAstronomical Geometry

Astronomy lends itself well to the use of spherical coordinates. For example, stars and

planets are spheres and many other sources are approximated as spheres. The sky itself

is a spherical ‘dome’, and photons from astrophysical sources typically escape in all

directions, passing through the imaginary surfaces of spheres. Thus, it is important to

understand this geometry.

B.1 One-dimensional and two-dimensional angles

The arc length of a circle, s, is related to the angle, u, via (see Figure B.1a),

s ¼ r u ðB:1Þ

for u in radians and s in the same units as r. Over an entire circle, u subtends 2p radians

so the full circumference of the circle is s ¼ 2p r. When measuring astronomical

sources from Earth, distances are so large that an arc length, s, and line segment, d, are

virtually indistinguishable for small u (Figure B.1a, Prob. 1.2) so Eq. (B.1) can be used

to determine the projected linear size of a source if the distance is known in most cases.

By ‘projected’, we mean projected onto the plane of the sky as shown, for example, in

Figure B.1b.

By analogy, the surface area of a section of a sphere, s, is related to the two-

dimensional solid angle, O, via,

s ¼ r2 O ðB:2Þ

Astrophysics: Decoding the Cosmos Judith A. Irwin# 2007 John Wiley & Sons, Inc. ISBN: 978 0-470-01305-2(HB) 978-0-470-01306-9(PB)

for O in units of steradians (sr) and s in the same units as r2. For a given surface area

(s constant), it is clear that the solid angle falls off as 1r2. Over an entire sphere, O

subtends 4p sr so the full surface area of the sphere is s ¼ 4p r2. Again, ifO is small, sapproximates the projected surface area of a real object on the sky.

For spherical sources which are circular in projection and subtend a small angle on

the sky, a useful expression relating the one-dimensional angle subtended by the

source, u, to its two-dimensional solid angle, O, is easily obtained. Since the area of

the circle on the sky is s ¼ pðd2Þ2 pðs

2Þ2

, combining this with Eqs. (B.1) and (B.2)

yields,

O ¼ p

4u2 ðB:3Þ

for u in radians and O in steradians. If the source in the sky is elliptical rather than

circular, in projection, then,

O ¼ p

4u1 u2 ðB:4Þ

where u1 and u2 are the angular major and minor axes, respectively, of the ellipse.

B.2 Solid angle and the spherical coordinate system

Cartesian (x; y; z) and spherical (r; u;f) coordinate systems are shown in Figure B.2

where the unit sphere is shown. The origin of the spherical coordinate system can, of

course, be placed wherever needed, for example, at the centre of a star, the surface of a

star, or at a detector on the Earth. The coordinate angle, u, is customarily measured

downwards from the positive z direction. If the origin is placed at the surface of the

Earth, then the positive z direction is towards the zenith and the angle, u, is called the

zenith angle. In spherical coordinates, an infinitesimal solid angle is,

dO ¼ sin u du df ðB:5Þ

(a) (b)

r dD

i

line segment d

arc S distance r

θ

Figure B.1. (a) For angles, u, that are very small (here exaggerated), the arc length, s, is equal tothe linear segment, d, in the plane of the sky. (b) Example of a non-spherical source and itsprojection in the plane of the sky. In this case, one dimension of a flat object such as a galaxyprojects into the sky plane such that d ¼ D cos i, where i is the angle of inclination.

344 APPENDIX B ASTRONOMICAL GEOMETRY

which relates the solid angle to the individual independent angles, u and f. Eq. (B.5) is

especially helpful in integrations over angles that are large. We can check to ensure that

an integration of dO over all angles yields 4p sr, for example,

ZOdO ¼

Z 2p

0

h Z p

0

sin u duidf ¼

Z 2p

0

h cos pþ cos 0

idf ¼

Z 2p

0

2 df ¼ 4p

ðB:6Þ

sinθ dφ

dφ

dθ θ

O

dφ

x

y

z

φ

dθ

Figure B.2. The spherical coordinate system (r; u;f) showing the unit sphere (r ¼ 1), the angles,u, f, and their infinitesimal increments, d u and d f, respectively. The origin is at O. Aninfinitesimal solid angle, dO is shown as the hatched region with sides of length, du and sin u df.

B.2 SOLID ANGLE AND THE SPHERICAL COORDINATE SYSTEM 345

Appendix CThe Hydrogen Atom

C.1 The hydrogen spectrum and principal quantum number

Hydrogen is the most abundant element in the Universe (see Sect. 3.3) and is also the

simplest, with a single proton for the nucleus and a single bound electron in its neutral,

non-isotopic form. A useful description of this atom can be obtained by first assuming

that the electron is in uniform circular orbit about the nucleus. The Coulomb force

holding the electron to the nucleus is thus balanced by the centripetal force of the

electron’s motion,

F ¼ e2

r2¼ me v

2

rðC:1Þ

where r is the separation between the electron and the proton, me is the electron mass, e

is the charge on the electron/proton, and v is the electron’s speed.

Classically, since any accelerating charged particle should radiate, one would expect

continuous emission from such an orbiting electron. However, this is not observed.

Instead, discrete emission lines at specific frequencies are seen. This puzzle led to the

development of quantum mechanics, the simplest approximation of which is the Bohr

model of the atom. In this model, the angular momentum of a bound electron, L, is

quantized, meaning that it can take only discrete, rather than continuous, values. The

electron would not radiate when in this allowed state. Radiation results only when an

electron makes a transition from one state to another. As strange as these assumptions

seem, they describe atoms and their observed spectra very well.

The quantized angular momentum is given by,

Ln ¼ me v rn ¼ nh

2pðC:2Þ


where h is Planck’s constant and n is an integer called the principal quantum number1.

Combining Eqs. (C.1) and (C.2) and solving for rn yields the quantized radius,

rn ¼ n2h2

4p2 me e2¼ n2 a0 ¼ n2 5:3 109 cm ðC:3Þ

where a0 is called the Bohr radius (Table G.2) and gives the size of the hydrogen atom

when n = 1, i.e. when the atom is in its ground state.

The energy of a bound electron is the sum of its kinetic and potential energies. For a

non-quantized case,

E ¼ Ek þ Ep ¼1

2me v

2 e2

rðC:4Þ

where we have used the usual convention that the potential energy is zero at a

separation of infinity. Putting Eqs. (C.2) and (C.3) into Eq. (C.4) yields the quantized

energy,

En ¼ 1

n2me e

4 2p2

h2¼ 2:18 1011

n2erg ¼ 13:6

n2eV ðC:5Þ

Thus the energy of a bound electron is negative. From Eqs. (C.3) and (C.5), it is clear

that quantization of angular momentum implies quantization of the radius and energy

as well.

When an electron makes a transition from a higher to lower energy state, then a

photon is emitted that carries away the excess energy. The energy difference between

states is,

DE ¼ En0 En ¼ 13:6 1

n02 1

n2

eV ðC:6Þ

where n0 is the higher energy quantum level and n is the lower level. If a transition is

from a higher to lower energy state (DE positive), then a photon is emitted and an

emission line is observed. If an atom is perturbed by a collision or a photon which has

the energy of an allowed transition, then this energy can be absorbed by the electron

and it will make a transition from a lower to a higher energy state (DE negative). An

absorption line may then be observed. More information as to when emission and

absorption lines are observed is given in Sect. 6.4.2.

Equating the energy difference of Eq. (C.6) to the energy of an emitted photon

DE ¼ h n ¼ h c=l, and evaluating the constants leads to a photon frequency and

wavelength, respectively, given by,

n ¼ 3:29 1015 1n2

1

n02

Hz

1

l¼ R

1n2

1

n02

cm1 ðC:7Þ

1The same equation results by expressing the electron mass in terms of its de Broglie wavelength (Table 1.1) andrequiring an integral number of wavelengths around a circular orbit, i.e. nl ¼ 2p rn.

348 APPENDIX C THE HYDROGEN ATOM

where R is the Rydberg constant for hydrogen,

R ¼ R1 M

Mþme

¼ 109 677:6 cm1 ðC:8Þ

where M is the mass of the nucleus (for hydrogen this is just the mass of a proton,

M ¼ mp) and R1 (Table G.2) is the Rydberg constant for an ‘infinitely’ heavy nucleus

in comparison to the electron.

Table C.1. Line data for the hydrogen atoma

Designation Transitionb i–j li;jðnmÞe fi; jc Aj;i

d(s1)

Lyaf 1–2 121:567 0:4162 4:699 108

Ly b 1–3 102:572 0:07910 5:575 107

Ly g 1–4 97:254 0:02899 1:278 107

Lylimit 1–1 91:18

Ha 2–3 656:280 0:6407 4:410 107

Hb 2–4 486:132 0:1193 8:419 106

H g 2–5 434:046 0:04467 2:530 106

H d 2–6 410:173 0:02209 9:732 105

H e 2–7 397:007 0:01270 4:389 105

H8 2–8 388:905 0:008036 2:215 105

Hlimit 2–1 364:6

Pa 3–4 1875:10 0:8421 8:986 106

P b 3–5 1281:81 0:1506 2:201 106

P g 3–6 1093:81 0:05584 7:783 105

Plimit 3–1 820:4

Ba 4–5 4051:20 1:038 2:699 106

Bb 4–6 2625:20 0:1793 7:711 105

B g 4–7 2165:50 0:06549 3:041 105

Blimit 4–1 1458:4

H109ag 110–109 5:985 cm 7:0 104

HI 1–1h 21:106114 cm 2:876 1015

Deuterium I 1–1h 91:5720 cm 4:65 1017

aFrom Ref. [44] unless otherwise indicated.bThe upper level is j and the lower level is i, where i and j are indices representing theprincipal quantum numbers, unless otherwise indicated.cAbsorption oscillator strengths (see Appendix D.1.3) for transitions from lower level i toupper level j.dAverage EinsteinA coefficient for transitions fromupper level j to lower level i. The averagevalue means that the particles are assumed distributed in their substates (determined by theorbital angular momentum quantum number, l) according to the statistical weights of thosesubstates.eUnits are nm unless otherwise indicated.fThe Einstein A coefficient for the permitted transition is A2p!1s ¼ 6:27 108 s1 (Ref.[50]). See Sect. 8.4 for a discussion of the forbidden transition.gThe wavelength is from Eq. (C.7) and the Einstein A coefficient from Anþ1!n ¼1:167 109=n6 Ref. [50].hTransition between the hyperfine splitting of the ground state.

C.1 THE HYDROGEN SPECTRUM AND PRINCIPAL QUANTUM NUMBER 349

Eq. (C.7) predicts the spectrum of the hydrogen atom, some lines of which are listed in

TableC.1 and an energy level diagramshowing a number of transitions is shown inFigure

C.1. Table C.1 also gives the oscillator strength (see Appendix D.1.3), as well as the

Einstein A coefficient for the transition. The Einstein A coefficient indicates the sponta-

neous downwards transition rate that is expected as a result of intrinsic properties of the

atom. The inverse of this quantity indicates the time that the electron can remain in the

higher energy level before spontaneously de-exciting in the absence of any external

perturber, either radiative or collisional. Series of transitions are named according to a

common lower energy level. For example, the Lyman series, designated Ly in Table C.1,

denotes transitions between the ground (n ¼ 1) state and all higher levels. The Balmer

series, denoted H, famous because the lines occur mainly in the visible part of the

spectrum, indicate transitions between n ¼ 2 and higher levels. Other series are the

Paschen (P, n ¼ 3) and Bracket (B, n ¼ 4) series. The transitions between closest prin-

cipal quantumnumbers is calleda, the next up,b, and so on. Thus,Ha corresponds to thetransition between n ¼ 3 and n ¼ 2. This Greek letter nomenclature continues to very

high energy states where the series is designated simply by H and the principal quantum

number of the common lower state. For example, H50a and H250b would designate

transitions between n ¼ 51 and n ¼ 50, and between n ¼ 252 and n ¼ 250, respectively.

Ground state

Lymanseries

Ionized atom(continuous energy levels)

Paschenseries

Balmerseries

Excitedstates

Ene

rgy

(eV

)

–15

–13.6

–10

–5

–3.4

–1.5

–0.85

E = 0

n = 2

n = 1

n = 3

n = 4n = 5

Figure C.1. Energy level diagram for hydrogen, showing the various series seen in the hydrogenspectrum.


At higher values of n, the energy levels become closer together (Figure C.1,

Table C.1, Eq. C.7) and transitions between more closely spaced energy levels result

in photons of lower energy and lower corresponding frequency. Thus, transitions such

as the H250a line will be in the radio part of the spectrum, the Brackett series results in

infrared emission, the Balmer series lines are in the optical and Lyman series in the

ultraviolet. The highest possible energy transition is between n ¼ 1 and n0 ¼ 1,

essentially the continuum rather than a bound state. From Eq. (C.6), it can be seen

that this energy is 13.6 eV. This is also, therefore, the energy required to ionize the

hydrogen atom. The threshold photoionization cross-section (i.e. the value applicable

to a photon with just sufficient energy for ionization) for hydrogen from a quantum

level, n is given by (Ref. [44]),

sH!HII ¼8 h3 g n

3ffiffiffi3

pp2 me

2 c e2¼ 7:907 1018 n g cm2 ðC:9Þ

where g is called theGaunt factor given in Table C.2. This value gives the effective area

that the hydrogen atom presents to an incoming ionizing photon.

Table C.2. Quantum numbers for the first five energy levels of hydrogen

Number of

na lb mlc ms

d Designatione possible statesf gng gh

1 0 0 1=2 1s 2 2 0:80

2 0 0 1=2 2s 2 8 0:89

1 1; 0; 1 1=2 2p 6

3 0 0 1=2 3s 2 18 0:921 1; 0; 1 1=2 3p 6

2 2:::2 1=2 3d 10

4 0 0 1=2 4s 2 32 0:941 1; 0; 1 1=2 4p 6

2 2:::2 1=2 4d 10

3 3:::3 1=2 4f 14

5 0 0 1=2 5s 2 50 0:951 1; 0; 1 1=2 5p 6

2 2:::2 1=2 5d 10

3 3:::3 1=2 5f 14

4 4:::4 1=2 5g 18

aPrincipal quantum number. This is an integer starting at 1.bOrbital angular momentum quantum number. l ¼ 0; 1 . . . ðn 1Þ, where ::: means ‘add 1’.cMagnetic quantum number, ml ¼ l . . .þ l.dSpin quantum number, ms ¼ 1=2.eDesignation of the state for a given l. The number specifies the principal quantum number and the letterdesignates the value of l, where the letters are s, p, d, f , and g for l ¼ 0; 1; 2; 3; and 4, respectively.fThe number of possible states for a given l. Since there are 2 lþ 1 values of ml and two possible spins for each,there are 2ð2 lþ 1Þ states for any l.gStatistical weight for the energy level. This is the number of possible states for a given n,gn ¼

Pn1l¼0 2ð2 lþ 1Þ ¼ 2 n2.

hGaunt factor for the level (Ref. [44])

C.1 THE HYDROGEN SPECTRUM AND PRINCIPAL QUANTUM NUMBER 351

C.2 Quantum numbers, degeneracy, and statistical weight

The Bohr model is adequate to explain the energy levels of the hydrogen atom and Eq.

(C.7) is a good predictor of the observed spectral lines listed in Table C.1. However,

deeper scrutiny reveals some fine structure that cannot be explained using the principal

quantum number alone. In addition, if the atom is placed within a magnetic field, the

energy levels split into sublevels. To explain this behaviour, a full quantummechanical

development is required in which four quantum numbers are needed to describe the

state of the electron: the principal quantum number, n, the orbital angular momentum

quantum number, l, the magnetic quantum number, ml, and the electron spin quantum

number, ms. These numbers and their relationships to each other are given for the first

five energy levels of hydrogen in Table C.2.

The orbital angular momentum of the electron is now given by,

L ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilðlþ 1Þ

p h

2pðC:10Þ

instead of Eq. (C.2) of the Bohr model. Rather than considering the electron to be

‘orbiting’ the nucleus, l gives the three-dimensional shape of a probability distribution

about the nucleus within which the electron can be found. The electron spin, which can

either be þ1=2 (‘up’) or 1=2 (‘down’), represents the angular momentum of the

electron alone. Although it may be helpful to visualise electron spin in the classical

sense like a spinning top, ms should more accurately be thought of simply as an

intrinsic property of the electron. The magnetic quantum number indicates the quan-

tized projection of the angular momentum vector along a preferred axis, z, the latter

being that of an externally applied magnetic field.

A set of quantum numbers that can be used instead of those of Table C.2 are n, l, j,

and mj, where n and l are the same as before, but j is the total angular momentum

quantum number and mj is the magnetic quantum number corresponding to j. The

quantum number, j, includes both orbital angular momentum and the electron’s spin

together, andmj is the projection of j onto the z axis2. This system does not contain any

new information beyond n, l, ml, and ms and can be derived from this set of quantum

numbers. However, it can be useful to consider the electron’s orbital angular momenum

and spin angular momenum together, especially when considering fine structure, as we

do in the next section.

Table C.2 shows that the electron can be in a number of possible states for any given

principal quantum number. The number of states at a given n is called the statistical

weight or the ‘degeneracy’ of the level and is given by,

gn ¼ 2 n2 ðC:11Þ

2The quantun number, j takes values j ¼ l 1=2, unless l ¼ 0 in which case j ¼ 1=2. The quantum number, mj

takes values from j to þj resulting in 2 jþ 1 values.


The Pauli Exclusion Principle states that no two electrons can have the same set of

quantum numbers. Thus, if there were two electrons in the atom, such as in the negative

hydrogen ion, H, which is found in abundance at the surface of the Sun, then they

cannot occupy the same state.

A spectral line is thus formed when the electron makes a transition from one set of

quantum numbers into another. However, not every possible change in quantum

number can occur or can occur with equal probability (see Ai;j of Table C.1). Certain

selection rules apply, for example, D l ¼ 1. Under certain circumstances, however,

highly forbidden lines can be seen, as we shall see in Sect. C.4. There are no selection

rules for D n, hence, all possible D n are permitted.

C.3 Fine structure and the Zeeman effect

Fine structure is the splitting of energy levels by a small amount due to relativistic

effects as well as the fact that electron spin couples with the orbital angular momentum,

forming a total angular momentum vector with quantum number, j. This leads to a

modification of the energy equation (Eq. C.5) by a small amount of order me c2 a4,

where a ¼ 1137

is the dimensionless fine structure constant. The result is a lowering

(more negative) of the energy of the n ¼ 1 level (j ¼ 1=2) by,

DE ¼ me c2 a4

8¼ 2 104 eV ðC:12Þ

and a lowering of the j ¼ 1=2 and j ¼ 3=2 states in the n ¼ 2 level by different amounts

resulting in a splitting of this level into two sublevels separated by,

DE ¼ me c2 a4

32¼ 4:5 105 eV ðC:13Þ

(Ref. [100]).

If the atom is placed in a magnetic field, B, with direction, z, the total angular

momentum vector precesses about the field direction, called classically Larmor pre-

cession, and there is a magnetic moment in the atom whose quantization is given by,

m ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijðjþ 1Þ

pmB, where mB ¼ e

2me ch2p

¼ 9:24 1021 ergG1 is called the Bohr

magneton. It is the magnetic quantum number, mj, that describes the projection of the

total angular momentum along the z axis. The energy state will split into sub-levels

whose separation depends on the strength of the field. If a transition between energy

states occurs such that Dmj ¼ 0 or Dmj ¼ 1, then, respectively, the resulting line will

be unshifted, or there will be a shift in the frequency of the spectral line according to,

n ¼ n0 mB

hB ¼ n0 1:4 106 B Hz ðC:14Þ

C.3 FINE STRUCTURE AND THE ZEEMAN EFFECT 353

where n0 is the frequency of the line in the absence of the field. In a gas in which many

such transitions are occurring, then three lines can be observed instead of one3. This is

called the Zeeman effect4 and, if measurable, is a very important way to obtain the

strength of the magnetic field in an object. Since interstellar magnetic fields have

strengths of order a few 106 G, a typical shift for interstellar lines is of order a few Hz.

Most of the lines given in Table C.1, have frequencies of 1014 to 1015 Hz, so this is a

negligible shift. However, if the magnetic field is stronger or a lower frequency line is

chosen, then the Zeeman effect becomes measurable. For example, the Zeeman effect

is the primary way in which we know the magnetic field in Sunspots which have strong

fields of order 1000 (D n ¼ 1:4 MHz) to 4000 (D n ¼ 5:6 MHz) G in comparison to a

globally averaged Solar field of about 1 G. Zeeman splitting has also been measured in

the HI l 21 cm line (see below) in regions in which the interstellar magnetic field is

stronger than average.5

C.4 The l 21 cm line of neutral hydrogen

There is one more effect that has not yet been considered, that of the intrinsic angular

momentum of the nucleus, called nuclear spin and designated I. For the normal

hydrogen atom in which the nucleus is a proton, I ¼ 1=2 and for deuterium, I ¼ 1.

Just as the electron’s orbital angular momentum and spin angular momentum could be

combined into a total angular momentum quantum number, j, so now the nuclear spin

can be included with the total angular momentum to form a total angular momentum

quantum number for the whole atom, called F. This coupling is only important for the

ground state of hydrogen where the electron is closest to the nucleus.

The result is hyperfine splitting of the ground state. For hydrogen, it is helpful to

think of this classically as either the proton and electron spinning in the same sense

(F ¼ 1), or the proton and electron spinning in opposite senses (F ¼ 0), the former

orientation having the higher energy. The statistical weights of these two states are

given by g ¼ 2F þ 1. Hence the spin-parallel state has a statistical weight of gu ¼ 3

and the spin-anti-parallel state has a statistical weight of gl ¼ 1, where the subscripts,

u, and l refer to the upper and lower energy levels, respectively.

The l 21 cm transition is sometimes called the spin-flip transition since de-excitation

involves this process. The transition from F ¼ 1 to F ¼ 0 is a forbidden transition

meaning, in this case, that it violates the D l ¼ 1 selection rule. This does not mean

that it cannot occur, only that it is extremely rare in comparison to other lines (see the

3The three lines are polarized with the unshifted line polarized in a different sense than the shifted lines. If thelines are viewed from an angle perpendicular to the magnetic field direction, then all three lines are observed, butif the viewing angle is along the field direction, then the two shifted lines are seen but not the unshifted line.4There is also an anomalous Zeeman effectwhich is line splitting by amounts different from Eq. (C.14). In generalEq. (C.14) must be modified to include a function of the various quantum numbers (called the Lande g factor). SeeRef. [143].5The Zeeman effect is not restricted to the hydrogen atom. It is regularly observed in other atoms as well as inmolecules.


Einstein coefficient in Table C.1). In very low density gas, some fraction of the

hydrogen will have time to spontaneously de-excite and emit a photon before inter-

acting with another particle. For most ISM densities, however, it is collisions between

particles that will cause the line to be excited or de-excited. Since there is so much

atomic hydrogen in space, it is possible to observe the resulting spectral line, the

l 21 cm line of hydrogen. This radio line is exceedingly important in astrophysics

providing us with an enormous amount of information, from galaxy dynamics to

cosmology, as is described further in Sect. 9.4.3. It is astonishing that one of the

most powerful probes of our Universe rests with a tiny, insignificant hyperfine energy

shift in the simplest of all atoms.

C.4 THE l 21 CM LINE OF NEUTRAL HYDROGEN 355

Appendix DScattering Processes

Scattering involves a brief interaction between light and matter that results in a change

of direction of incoming photons, as Figure 5.1.a illustrates. The observed light

intensity will diminish because only a fraction of the incoming photons are scattered

into the same line of sight as the original beam. If the light is scattered with equal

probability into all 4 sr, then the scattering is isotropic1. If the outgoing photons have

the same energy as the incoming photons, then the scattering is said to be elastic and, if

some energy is lost, the scattering is inelastic, similar to the treatment of colliding

particles in the discipline of Kinematics.

To understand some scattering processes, however, it is helpful to treat the signal as a

wave because the scatterer may be a charged particle that is responding to the

oscillating electric field of the incoming light. Elastic scattering is called coherent

scattering2 in this model, with the outgoing energy, and therefore wavelength

(E ¼ h c=), of the scattered wave remaining unchanged. Inelastic scattering then

corresponds to incoherent scattering.

In either approach, the result is the same and the fraction of light that is lost from the

line of sight is directly proportional to the particle’s effective scattering cross-section,

s. It is therefore of some importance to determine s for different types of particles andscattering circumstances. A few of these, important to astrophysics, will now be

described.


1In reality, scattering may not be isotropic, but is often symmetric in the forward/reverse directions.A mathematical function that describes the strength of the scattering with angle is called the phase function.2‘Coherent’ classically implies that the phase of the outgoing light is also the same as that of the incoming beam.This is true for coherent scattering except in the case of resonance (see Appendix D.1.4 and Ref. [77]).

D.1 Elastic, or coherent scattering

D.1.1 Scattering from free electrons – Thomson scattering

An example of elastic scattering is Thomson scattering which occurs when light

scatters from free electrons. For the photon to be scattered elastically, its energy

must be much less than the rest mass energy of the electron (otherwise some of the

energy is lost to the electron, as will be described in Appendix D.2), i.e.,

Eph me c2 ¼ 8:2 107 erg ¼ 0:51 MeV ðD:1Þ

where me is the mass of the electron and c is the speed of light. Since this energy occurs

in the wave part of the spectrum (Table G.6), Thomson scattering can occur for any

light frequency up to at least soft X-rays before energy exchange starts to become

important.When thewave impinges on an electron, thewave’s oscillating electric field,~E, induces an oscillation of the electron at the same frequency due to the electric field

component of the Lorentz force,~F ¼ e~E (see Table I.1), where e is the charge on the

electron. An oscillatory motion is also an accelerating motion, and any accelerating

charged particle, in this case an electron, will radiate. The result is electric dipole

radiation. The electron can be thought of as absorbing a wave and re-emitting it again

at the same frequency but over a wide range of angles which, for Thomson scattering, is

symmetric in the forwards and backwards directions with less emission towards the

sides. The electron oscillates in phase with the incoming wave, and no energy is lost as

a result of the interaction.

Note that the oscillating magnetic field of the incoming wave could also have

initiated an oscillation of the electron via the magnetic component of the Lorentz

force,~FB ¼ e ~vc~B (Table I.1). However, for electromagnetic radiation, the magnitude

of the electric field vector equals the magnitude of the magnetic field vector, j~Ej ¼ j~Bj,in cgs units. In order for the magnetic oscillation to have the same amplitude as that

induced by the electric field, the electron velocity would have to equal the velocity of

light. The oscillating magnetic field can therefore be neglected for Thomson scattering.

The probability of a photon interactingwith an electron is determined by theThomson

cross-section, which is the area presented by the electron to an incoming photon3,

T ¼ 8

3r2e ¼

8 e4

3m2e c

4¼ 6:65 1025 cm2 ðD:2Þ

where re is the classical electron radius, given in Table G.2. Note that this cross-section

is independent of wavelength so the fraction of scattered emission compared to

incident emission is the same for all wavelengths. Eq. (D.2) would also apply to a

free proton with me replaced by the proton mass, mp, but it is clear that the proton cross

section would lower by a factor of ðme=mpÞ2 107 and therefore negligible in

3The ‘effective area’ for an interaction may not be exactly the ‘geometrical’ area (see Sect. 3.4.3).

358 APPENDIX D SCATTERING PROCESSES

comparison to the electron cross-section. Therefore, it is the free electrons in an ionized

gas that cause the scattering, rather than the free protons.

If the incoming signal is unpolarized, composed of many plane waves with a random

orientation of~E perpendicular to the direction of propagation, then the scattered waves

froma distribution of free electronswill be polarized in a direction 90 from the direction

of the incoming signal. This is because an oscillating electron can only produce a

transverse wave, as shown in Figure D.1. Thus, Thomson scattering polarizes light.

D.1.2 Scattering from bound electrons I: the oscillator model

Scattering from bound electrons is a more complex situation (see Ref. [143] or Ref.

[79] for details) since there are potentially many scatterers (many bound electrons) and

many natural frequencies in the atom or molecule (many possible quantum transitions).

This problem has been approached by first treating a single bound electron as a

classical harmonic oscillator like a mass on a spring, and adjusting the results as

appropriate to a real quantum mechanical system later. The model is motivated by the

fact that, as with Thomson scattering, the electron should oscillate in response to the

impinging oscillating electromagnetic wave, yet the electron is also anchored, via a

Coulomb force, to its parent atom or molecule like a mass would be anchored via a

connecting spring to a wall.

Just like a mass on a spring, a single bound electron will oscillate at a natural

frequency if it is perturbed from its equilibrium configuration. This natural frequency

can be expressed either as an angular frequency, !0, or a frequency, 0, where!0 ¼ 2 0. Since the electron is oscillating (and therefore accelerating), it also emits

radiation. This emitted radiation provides a radiation reaction force which damps the

oscillations so that the electron eventually settles down into its equilibrium state again.

Thus, in response to a single perturbation, the electron behaves as a damped harmonic

oscillator. The damping is exponential with time, t, such that, averaged over a single

oscillation, the energy declines as ecl=t. Here, cl is called the classical damping

constant,

cl !02 e ¼ ð2Þ2 20 e ðD:3Þ

where,

e 2 e2

3me c3¼ 6:3 1024 s re

cðD:4Þ

is approximately the time for light to travel a distance equal to the classical electron

radius, re. Therefore, the classical damping constant is,

cl ¼82 e2 203me c3

¼ 2:47 1022 20 s1 ðD:5Þ

D.1 ELASTIC, OR COHERENT SCATTERING 359

(a) (b)

(c) y Observer sees unpolarised light

incoming unpolarised light

cloud of scattering electrons

Scatteringelectron

Scatteringelectron

E Scattered

E Scattered

E Scattered

E of incoming wave

incoming light

Observer sees polarised light

Observer sees polarisedlight



x

z

E Scattered

E Scattered

E Scattered

E of incoming wave

incoming light

Figure D.1. Geometry showing how unpolarized light can be polarized as a result of Thomsonscattering or Rayleigh scattering. The three-dimensional x y z axis is shown at lower left and,in each panel, the incoming light is propagating in the positive x direction with its electric fieldvector,~E, perpendicular to the direction of propagation. The electron will oscillate in the directionof oscillating ~E, emitting a scattered electromagnetic wave. (a) In this case, ~E of the incomingwave, and therefore the motion of the electron, oscillates in the z direction. The outgoingscattered radiation due to an oscillation in z will be only in the x y plane. Three of the scattered~Evectors are shown on the positive x and positive and negative y axes but they could have beendrawn at any point in the x y plane. (b) In this case, the incoming wave’s ~E lies in the ydirection, resulting in scattered radiation in the x z plane only. Three of the scattered~E vectorsare shown. (c) This case illustrates the situation in which there are many incoming waves beingscattered off many electrons. The incoming wave is unpolarized, meaning that there are many ~Evectors all of which are perpendicular to the direction of propagation, though only the two in the y and z directions are shown. An observer looking back along the direction of propagation willcontinue to see unpolarized scattered light. However, any observer perpendicular to the direction ofpropagation would see a polarized signal, as illustrated.


The lifetime of the oscillation, is called the damping time,

damp ¼ 1=cl ðD:6Þ

An oscillation corresponding to an optical light frequency of 0 6 1014 Hz, for

example, would give a damping time of damp 10 ns. This is much longer than the

period of oscillation itself, T0 ¼ 1=0 1015 s so there will be many oscillations

before the energy is dissipated. The case of the damped harmonic oscillator is the

classical analogue of spontaneous line emission, a subject towhich wewill return in the

next section (Appendix D.1.4).

If the perturber is now an incident electromagnetic wave, then instead of a single

perturbation, the electron is continuously forced by the applied oscillating electric field

of the wave. It now behaves as a forced, damped harmonic oscillator. The resulting

scattering cross-section in this harmonic oscillator model is a function of frequency,

sð!Þ ¼ T

!4

ð!2 !20Þ

2 þ ð!03 eÞ2

ðD:7Þ

where ! ¼ 2 is the angular frequency of the perturbing wave. The forcing fre-

quency may be similar to, higher, or lower than the natural frequency, and the type of

scattering that results depends on which regime is being considered. These are:

ðaÞ ! !0:

If the incoming wave’s frequency is much greater than the natural frequency then

Eq. (D.7) reduces to the constant,

s ¼ T ðD:8Þ

so the situation approximates that of Thomson scattering. This is because the period of

the natural oscillation, T0 ¼ 1=ð2!0Þ, is much greater than the forcing time

T ¼ 1=ð2!Þ, of the incident radiation. The bound electron does not have time to

oscillate naturally in the presence of such rapid forcing so the electron simply responds

to the external force as if it were ’free’ of its natural binding to the atom.

ðbÞ ! !0:

If ! !0, then ð!2 !20Þ ð! !0Þð!þ !0Þ ð! !0Þ 2!0 and Eq. (D.7)

becomes,

sð!Þ ¼T

2 e

ðcl=2Þð! !0Þ2 þ ðcl=2Þ2

ðD:9Þ

where we have used Eq. (D.3). This describes resonance scattering and occurs when

the incoming photon has a frequency approximately equal to the frequency at which the


system would naturally oscillate. The function has a peak4 at !0 and declines as !departs from !0. This resonance is similar to other common systems in which a forcing

frequency is similar to a natural frequency, such as pushing a child on a swing. The

scattering cross-section achieves its highest values for this case.

ðcÞ ! !0:

In this case, the perturbing electric field appears ‘frozen’ in comparison to the rapid

natural oscillation of the electron, so the perturbing force acts like a static force. Eq.

(D.7) then becomes (noting that, from Eq. (D.3), !30e ¼ !0cl !2

0Þ

sð!Þ ¼ T

!

!0

4

¼ T

0

4

ðD:10Þ

where we have used ! ¼ 2 c= and !0 ¼ 2 c=0. This is called Rayleigh scattering

and has a strong dependence on wavelength, .

D.1.3 Scattering from bound electrons II: quantum mechanics

We now wish to extend the classical oscillator model into the realm of quantum

mechanics. Classically, the accelerating, oscillating electron radiates and this radiation

produces the damping. Quantum mechanically, the ‘natural frequencies’ in the atom are

identified with the transitions between quantum states, so that 0 becomes i;j, i and j

representing the lower and higher energy states, respectively. Since an atom has many

natural frequencies that correspond to many different electron transitions governed by

quantum mechanical rules, the oscillator model must also be modified to take this into

account. This is achieved via a correction factor called the absorption oscillator strength,

fi;j. The oscillator strength is a unitless parameter that can be thought of as the number of

oscillators (electrons) that respond to incident radiation at frequency, . Values of fi;j forthe hydrogen atom and corresponding transition frequencies can be found in Table C.1.

With these modifications and expressing the result as a function of instead of!, cases a,b, and c from the previous section become respectively

s ¼ T cm2 i;j ðD:11Þ

sðÞ ¼ e2

me cfi;j LðÞ cm2 i;j ðD:12Þ

sðÞ ¼ T fi;j

i;j

4

cm2 i;j ðD:13Þ

4We have neglected a frequency shift in the peak of the line which is negligible in this classical development butcan be important in a quantum mechanical treatment. It is called the Lamb shift.


where,

LðÞ ¼1

i;j=ð4Þð i;jÞ2 þ ½i; j=ð4Þ2

Hz1 ðD:14Þ

i;j ¼ i þ j Hz ðD:15Þ

and

i ¼Xk<i

Ai;k j ¼Xk<j

Aj;k ðD:16Þ

We have also used Eqs. (D.2) and (D.5). Ai;k, Aj;k, are the Einstein A coefficients

between the lower state, i and all states, k, below it, and the upper state, j, and all states,

k, below it, respectively (Refs. [68], [77]), i.e. the spontaneous de-excitation rates for

the two energy levels5. These equations are equivalent to those expressed earlier for the

classical oscillator model except for the introduction of the oscillator strength and the

modification of the damping constant from its classical value, cl. Values of absorption

oscillator strengths and Einstein A coefficients for hydrogen can be found in Table C.1.

D.1.4 Scattering from bound electrons III: resonance scatteringand the natural line shape

For resonance scattering (Eq. D.12), each photon that is absorbed from a lower to an

upper state will result in an emission again from the upper to the initial lower state. The

scattering cross-section is then proportional to the function,LðÞ given by Eq. (D.14),which is called the Lorentz profile. It has a peak, Lði; jÞ, and a full width at half

maximum (FWHM), L, of, respectively,

Lði; jÞ ¼4

i; jL ¼ i; j

ð2Þ ðD:17Þ

where i;j is given by Eqs. (D.15) and (D.16), and is normalized such that the

integration over all frequencies (unitless) is equal to 1,

Z 1

0

LðÞ d ¼ 1 ðD:18Þ

5Eqs. (D.16) refer to spontaneous emission only, in the absence of any further perturbation. If the incidentradiation field is strong, then two more terms must be added: one to account for the absorption of photons whilethe particle is in the upper state resulting in transitions to higher energy levels, and one to account for stimulatedemission which is a process by which the incident photon induces an electron in the upper state to de-excite to thelower state without the absorption of the incident photon.


An example is shown in Figure D.2 and illustrates that this case is not perfectly

coherent since there is some probability that the photon may be emitted at a slightly

different frequency than it was absorbed, the probability distribution being described

by the Lorentz profile.

The damping constant, i;j, has a value that is somewhat larger than the classical

damping constant for an oscillator and takes into account the ‘fuzziness’ of each of the

upper and lower energy states. Using the Energy–Time form of the Heisenberg Uncer-

tainty Principle, the relation between photon energy and frequency (both Table I.1), and

the value of the line with, , from the Lorentz profile (Eq. D.17) we have, for a

transition between states, i, and j,

Ei;j ¼h

2

1

ti;j¼ h i;j ¼ h

i;j

2ðD:19Þ

Figure D.2. The Lorentz profile, LðÞ, plotted as a function of ( i; j) for the Ly line, usingthe classical damping constant of Eq. (D.5).


Thus, the quantum mechanical damping constant is associated with the uncertainty in

time for the transition,

i;j ¼1

ti;jðD:20Þ

The lifetime of an electron in an upper state, j, meaning the time that the electron would

remain in state j before spontaneously de-exciting to any lower state is,

j ¼1

j

¼ 1Pk<j Aj;k

s ðD:21Þ

where we have used Eq. (D.16), and the lifetime of an electron in upper state, j, before

making a transition to a specific lower state, i, is (Ref. [96]),

j;i ¼1

Aj;i¼ 1

3cl fj;is ðD:22Þ

Eq. (D.22) allows us to see the relationship between the Einstein A coefficient (the

de-excitation rate from state j to i) and the classical damping constant, cl, given by

Eq. (D.5) (but substituting i;j for 0). Note that the emission oscillator strength, fj;i isa quantity whose magnitude will not, in general, be the same as that of the absorption

oscillator strength, fi;j, because the number of possible states (i.e. the number of

possible ways that an electron can ‘fit’ into any given quantum level) is not

necessarily the same for the two levels. The number of possible states, i.e. the

statistical weight, must therefore be taken into account. The two oscillator strengths

are related via,

gj fj;i ¼ gi fi;j ðD:23Þ

where gj is the statistical weight of level, j, and gi is the statistical weight of level, i.

Values of absorption oscillator strengths and statistical weights for the hydrogen atom

are provided in Table C.1.

The process of resonance scattering involves absorption followed by re-emission by

bound–bound electrons. If we consider absorption and re-emission separately, the

Lorentz profile also describes the frequency response of the quantum mechanical

system for each process individually. For any spontaneous de-excitation, for example,

a bound–bound emitting transition (upper level j to lower level i) in an atom is

described by the Lorentz profile which is also called the natural line shape. The

Lorentzian line width, i;j=ð2Þ (Eq. D.17), is also called the natural line width or thedamping width and it is the width of a spectral line in the absence of any other line

broadening mechanisms. As described in Sect. 9.3, other effects will always broaden

the line to a value greater than the natural line width.


D.1.5 Scattering from bound electrons IV: Rayleigh scattering

The Rayleigh scattering equation (Eq. D.13) is equivalent to the result from the

classical oscillator model except for the presence of the absorption oscillator strength,

fi; j. Since there are various transitions, i; j, that are possible in an atom or molecule,

there should be many different regimes in which i; j, depending on the transition,i; j, being considered. However, it is possible to identify sets of lines such that i; jfor all of them. A good example is Rayleigh scattering in the Earth’s atmospherewhose

prominent lines occur mostly in the UV (e.g. for N2, O2 and H2O) whereas the Solar

spectrum peaks in the optical. With this in mind, the total quantum mechanical

Rayleigh scattering cross-section can be determined by considering a sum over the

various lines. The result is (Ref. [96]),

R T

Xk

f1;k 1;k2

2 112

cm2 ðD:24Þ

where f1;k is the absorption oscillator strength from quantum level 1 to k. Note that this

equation still gives R / 4 / 1=4, just like Eqs. (D.10) and (D.13).

Rayleigh scattering is a good example of how a physical phenomenon can be

approached in more than one way, as described in the introductory comments of

Chapter 5. Other than this ‘bottom-up’ scenario of counting oscillators in atoms and

molecules, the same result can be achieved by considering a molecule as a single

particle with some charge distribution. The regime corresponding to i;j (orequivalently i;j), now becomes >> the particle size6. If the molecule has a

net separation of charge (such as in the H2Omolecule) it is said to be an electric dipole.

However, incoming light with an oscillating electric field,~E, can induce an oscillatingelectric dipole moment,~p (see Table I.1), even in molecules that do not normally have a

permanent electric dipole moment, according to ~p ¼ ~E where is called the

polarizability of the molecule and is a measure of how easy it is to distort the charge

distribution. Thus, this scattering can be thought of as the response of thewhole particle

to the incoming oscillating electric field. Another approach is to consider all particles

together in a gas, in which case Rayleigh scattering is due to fluctuations or inhomo-

geneities in the dielectric constant of the gas, the latter being a measure of how easily a

material responds to an impinging electric field7. The dielectric constant, in turn, is

related to the index of refraction, n, of the material. Thus, R is sometimes written in

terms of n or in terms of the polarizability, . These approaches are essentially

equivalent8 and give the same result as the oscillator model, in particular, the 1=4

6Recall the simple case of the Bohr atom in which the particle size is related to the quantized wavelength,n ¼ 2 rn (see Appendix C, first footnote).7The dielectric constant is the ratio of the permittivity, , of the material in comparison to the permittivity of freespace, 0. Recall (Table I.1) that 0 ¼ 1=ð4Þ in cgs units.8See Ref. [121], for example, which shows the relationship between the index of refraction and oscillatorstrengths.


dependence of R. For mathematical expressions involving these other quantities, see

Ref. [96].

Rayleigh scattering is coherent and results in a scattered wave whose angular

distribution is the same as for Thomson scattering, that is, forward–backwards sym-

metric. Figure D.1, introduced to explain the polarization caused by Thomson scatter-

ing, also applies to Rayleigh scattering.

D.2 Inelastic scattering – Compton scattering from freeelectrons

Inelastic scattering occurs when a scattered photon loses some fraction (< 1) of its

energy. The outgoing scattered photon will then have a longer wavelength because of

the loss of energy. There are several examples of this (see Sect. 5.1.2), but here we

consider only Compton scattering which occurs when an incoming photon has

approximately as much or more energy as the rest mass energy of a free electron,

EphZme c2 ¼ 8:2 107 erg ¼ 0:51 MeV ðD:25Þ

If so, then some of the photon’s energy will be imparted to the electron on ‘impact’. The

situation is shown in Figure D.3 for an electron initially at rest. In this case, conserva-

tion of energy and momentum yields,

E0ph ¼

Eph

1þ Eph

me c2ð1 cos Þ

ð 6¼ 0Þ ðD:26Þ

where Eph ¼ h c= is the energy of the incident photon, E0ph ¼ h c=0 is the energy of

the scattered photon, and is the scattering angle. For low photon energies

Figure D.3. Geometry for Compton scattering when the electron is initially at rest.

D.2 INELASTIC SCATTERING – COMPTON SCATTERING FROM FREE ELECTRONS 367

(Eph me c2), this expression reduces to E0

ph ¼ Eph which is just elastic, or Thomson

scattering (Appendix D.1.1); the scattered photon has the same wavelength as the

incident photon and the electron remains at rest. For very high photon energies

(Eph me c2) and large scattering angles (cos 0), E0

ph me c2 so the scattered

photon is of order the rest mass energy of the electron, regardless of the energy (or

wavelength) of the incident photon. The energy imparted to the electron after impact

will be Ee ¼ Eph E0ph.

The net result is the ‘cooling’ of the radiation field as radiative energy is transferred

into kinetic energy of the electrons. The scattered photons will be redshifted with an

increase of wavelength,

¼ 0 ¼ Cð1 cos Þ ðD:27Þ

where,

C ¼ h

me c¼ 2:426 1010 cm ðD:28Þ

is the Compton wavelength for electrons. The Compton wavelength provides a value

for the change in wavelength after a single scattering (Prob 5.7).

For Compton scattering, the scattering cross-section is no longer the Thomson cross-

section, but rather the Klein–Nishina cross-section, KN, which includes quantum and

relativistic effects. The Klein–Nishina cross-section is smaller than the Thomson

cross-section (Figure D.4) and therefore scattering becomes less efficient as photon

energy increases.

Since Compton scattering occurs in regions of high energy photons, it is possible for

electrons in such regions to have high energies as well. If so, their velocities, which

could be relativistic, cannot be ignored. Conservation of energy and momentum,

therefore, must be applied in the centre-of-momentum frame of reference to derive

the general solution. We do not pursue this approach here, but the general solution

includes the possibility of energy transfer from the electron to the photon, a situation

that is known as inverse Compton scattering and is dealt with in greater detail in

Sect. 8.6. For an initially thermal electron (i.e. the electron velocity distribution is

Maxwellian), the condition for determining which direction energy is transferred is

given by (Ref. [101]),

h > 4 k Te Compton Effect

ðenergy transferred to electronsÞh ¼ 4 k Te ðno energy exchangeÞh < 4 k Te Inverse Compton Effect

ðenergy transferred to photonsÞ

ðD:29Þ

where Eph ¼ h is the initial energy of the photon, Te is the electron temperature, and k

is Boltzmann’s constant.


D.3 Scattering by dust

The interaction of light with dust can be complicated, but it is possible to gain some

insight into the interaction by starting with some simplifying assumptions, such as

grains that are spherical and of uniform composition. Dust scattering is not normally

considered in isolation, however, but rather is treated together with absorption, the total

being referred to as extinction (Chapter 5). The scattering and absorption properties

resulting from different grain compositions are taken into account via a complex index

of refraction (see below). A theory that takes this approach is called Mie Theory.

The extinction is expressed in terms of the unitless extinction efficiency factor for a

grain of radius, a,

Qext ¼ Cext

a2¼ Qabs þ Qscat ¼ Cabs

a2þ Cscat

a2ðD:30Þ

.5e–1

.1

.5

1.

.1e–1 .1 1. .1e2 .1e3

σ KN/σ

T

hν/mec2

Figure D.4. Ratio of the Klein–Nishina cross-section to the Thomson cross-section, KN=T as afunction of the parameter, h =me c

2, showing the decrease in cross-section with increasing photonenergy.

D.3 SCATTERING BY DUST 369

where Cext, Cabs and Cscat are the effective extinction, absorption and scattering

cross-sections, respectively (equivalent to eff , e.g. Sect. 3.4.3, Sect. 5.5) andQabs and

Qscat are the absorption and scattering efficiency factors, respectively. Thus, values of

Qmeasure the effective grain cross-section in comparison to its physical cross-section.

The albedo, that is, the fraction of incoming light that is scattered, first introduced in

Sect. 4.2, can also be expressed in terms of these quantities,

A ¼ Qscat

Qext

ðD:31Þ

The response of a grain to incoming radiation depends on its composition which can be

expressed in terms of a complex index of refraction, m, that has a real part, n, and an

imaginary part, k,

m ¼ n k i ðD:32Þ

where i ffiffiffiffiffiffiffi1

p. This expression is a mathematical convenience that allows one to

consider, together, the scattered portion of the wave, taken into account by n, and

the absorbed portion, taken into account by k. The scattered portion includes all

radiation that is not absorbed. This could include radiation that travels through the

grain and emerges out the other side, if this is occurring, radiation that is

diffracted about the grain, as well as radiation that is scattered in a more classical

sense, like a photon hitting a large ball. A pure dielectric material (a substance

with negligible conductivity) has k ¼ 0. Examples are ices and silicates

(Sect. 3.5.2). On the other hand, metals can have values of k and n that are

comparable in magnitude.

With these properties in mind, it is then possible to calculate the efficiency factors,Q

as a function of the size of the grain, a, and wavelength of incident light, . Both a and are usually folded into a single variable, x,

x 2 a

ðD:33Þ

and then plotted. Results for a weakly absorbing grain are shown in Figure D.5. Other

compositions show different curves, but the general trend of a strong increase, a peak,

and an approximately constant portion is quite typical.

The variable, x, is a useful parameter to plot because it indicates how efficiently

the grain interacts with incoming radiation in terms of the size-to-wavelength

ratio. It is clear that the strongest interaction (peak of the curve) corresponds to

a situation in which is of order the grain size. For the example given in

Figure D.5, the peak occurs at x 4, or a wavelength about 80 per cent of the

grain diameter.


If the grain size is very small in comparison to the wavelength of light (left on the

plot), then the extinction is a strong function of wavelength. In this case, the following

useful approximations result (Ref. [180]),

Qscat 8

3

2 a

4 m2 1

m2 þ 2

2 ðD:34Þ

Qabs 8 a

Imnm2 1

m2 þ 2

oðD:35Þ

where Im refers to the imaginary part of the term within the braces. As long as the grain

is not strongly absorbing (k is small), the term, ðm2 1Þ=ðm2 þ 2Þ is roughly constantwith wavelength or only weakly dependent on wavelength. Thus, for a given grain size

which is small in comparison to , Qscat / 1=4 (dash-dotted curve at left) which is

just Rayleigh scattering (Sect. D.1.5). For the absorption portion, Qabs / 1= (dotted

curve at left). In both cases, if the wavelength is very large in comparison to the grain,

then the interaction of the grain with light (Qext) is negligible, such as when radiowaves

traverse the ISM.

On the other hand, if the grain size is large in comparison to the wavelength of light

(right on the plot), then the scattering and absorption of radiation achieve a roughly

constant value, similar to light hitting a large solid surface. In this case, the scattering

and absorption cross-sections are just the geometric cross-sections (Qscat Qabs 1).

In this limit, the extinction efficiency factor Qext 2. This means that the effective

cross-sectional area of the grain is twice the geometrical cross-section, or twice as

much energy is removed from the incident beam by the particle than its geometrical

size would suggest. The reason for this is that light diffracts around the grain and this

bending also removes light from the line of sight.

Figure D.5. Absorption, scattering, and extinction efficiency factors as a function of theparameter, x ¼ 2 a=, for a weakly absorbing grain with index of refraction as indicated. When Qis multiplied by the geometric cross-section, the result is the effective interaction cross-section ofthe dust particle to the incident light. (Whittet, D.C.B, Dust in the Galactic Environment, 2nd Ed.,Institute of Physics Publishing)

D.3 SCATTERING BY DUST 371

The term Mie scattering is sometimes used to apply only to scattering for which a

particle is not small with respect to the wavelength (to right in Figure D.5), in order to

differentiate it fromRayleigh scattering (particles small with respect to thewavelength,

far left in figure). Since Qscat is about constant with respect to x at the right of

Figure D.5 then Qscat / 1= for a fixed particle size. This is a ‘greyer’ (flatter)

response with wavelength than Rayleigh scattering (/ 1=4) and is why larger

particles such as water droplets in clouds result in a greyer colour in comparison to

the blue sky (Sect. 5.1.1.3).

Although Mie theory is a simple approximation to the reality of complex grains,

more sophisticated theory indicates that, provided no grain alignment is occurring,

large modifications of the theory are not required (Ref. [160]). In reality, however, we

do know that some fraction of the grains must be aligned because scattering of starlight

by dust results in polarized light. In the interstellar medium, the degree of polarization,

Dp 2 per cent (Sect. 1.7). Polarization is generally not seen unless there is a magnetic

field, indicating that the magnetic field is playing an important role in aligning weakly

charged grains. The grain alignment mechanism is not yet entirely understood. It is

believed, though, that stellar radiation pressure, which exerts strong torques on the

grains, plays a part (Ref. [52]). The degree of polarization depends on the properties of

the grains and their orientation, as well as the wavelength of the incident light, so

studying the polarization of optical light as a function of wavelength is also a way of

obtaining information about the grains. Figure D.6 illustrates a possible configuration

that would result in polarized light.

Observer

Transmitted(optical)

Dust grain

Unpolarizedstar radiation

(optical)

IR dust radiation

Ω

B

z

x y

Figure D.6. An unpolarized signal from a background star becomes polarized as it interacts withan elongated dust grain that rotates about a magnetic field line. In this example, the magnetic fieldis parallel to the z axis and light propagates in the x direction. Incident light that has ~E in they direction is more likely to be absorbed than waves that have~E in the z direction, letting the lightthat has a z orientation pass through more easily. Thus, the transmitted light is polarized in a planealigned with the magnetic field. A charged dust grain that absorbs some fraction of incident lightwould also re-emit it in the infrared. The emitted light would be polarized in a plane perpendicularto the magnetic field


Appendix EPlasmas, the Plasma Frequency,and Plasma Waves

As indicated in Sect. 3.4, a plasma is an ionized gas. Under most astrophysical

conditions, a plasma is globally neutral1 since the negative charges from free electrons

will balance the positive charges of the ions. However, since the electrons are free, they

can be easily perturbed by an applied electric field. Figure E.1 illustrates such a

situation inwhich a small perturbation has caused some fraction of the charge to separate

in the x direction. Since there is a net positive charge onone side and a net negative charge

on the other, the plasma approximates a parallel plate capacitor with a region of width, x,

on either side corresponding to the two ‘plates’ of the capacitor, and globally neutral

ionizedmaterial is in between. Themagnitude of the electric field in this capacitorwill be

(Table I.1),

E ¼ 4p eNe

AðE:1Þ

where e is the charge on the electron, Ne is the number of electrons in a plate and A is

the area of a plate. The total number of electrons in a plate is,

Ne ¼ ne x A ðE:2Þ

where ne is the number density of electrons, taken to be roughly the same in all regions

since the perturbation is small. Therefore,

E ¼ 4p e ne x ðE:3Þ

1There are some exceptions, such as near sunspots or other regions in which there are ordered magnetic fields.There may be current flow in such circumstances.

The force on an electron produced by this field is (Table I.1),

F ¼ me €x ¼ e E ðE:4Þ

where me is the electron mass. Eqs. (E.3) and (E.4) lead to an equation for simple

harmonic motion, like a mass on a spring, i.e.,

€x ¼ v2p x ðE:5Þ

where

vp 4p e2 ne

me

1=2¼ 5:6 104 n1=2e rads s1 ðE:6Þ

is the angular plasma frequency. Thus, when electrons in a plasma are perturbed they

oscillate at a natural frequency, vp, similar to the way in which a bell rings at a natural

frequency when hit2. These oscillations are called plasma waves. One could consider

the response of the ions to such a perturbation as well, but the ions are very sluggish in

comparison to the electrons and can be considered at rest (Prob. 5.11).

Since accelerating electrons radiate, electrons in an ionized gas that oscillate at the

plasma frequency will themselves emit radiation at the same frequency. This is the

explanation for Type III solar radio bursts, as shown in Figure E.2. The declining

electron density between the Sun and the Earth3 corresponds to a declining plasma

frequency.

Since the response of the gas is as an harmonic oscillator, this development is similar

to that of the oscillator model for bound electrons described in Appendix D.1.2. That is,

we now consider the perturber to be an electromagnetic wave whose electric field4

+

+

+

+

–

+

–

+

+

–

+

–

–

+

–

+

+

–

+

–

–

+

–

+

–

–

–

–

× E ×

Figure E.1. Perturbation of the electric field in an astrophysical plasma induces a slight chargeseparation in it. The plasma then approximates a parallel plate capacitor filled with ionized, neutralmaterial within the ‘plates’, and having an electric field, ~E between them.

2Harmonics of this frequency may also be excited.3The electron density, ne, ranges from 108 cm3 in the Solar corona (see Figure 6.5) to 5 cm3 near the Earth(with some variation, Ref. [155a]).4As indicated in Appendix D.1, the oscillating magnetic field has a small effect in comparison to the oscillatingelectric field and can be neglected.

374 APPENDIX E PLASMAS, THE PLASMA FREQUENCY, AND PLASMA WAVES

oscillates at an angular frequency ofv, and consider what the result will be for differentregimes of v in comparison to vp (see below).

As the wave travels through a plasma, it will have a phase velocity (the rate at

which the peaks and troughs of the wave, or the phase, advances through the

medium),

nph ¼ c1

vp

v

21=2

ðE:7Þ

Figure E.2. This very low frequency radio emission resulting from a Solar flare was detected bythe Cassini spacecraft in Oct, 2003. A solar flare has produced a fast stream of highly energetic(1100 keV) electrons that encounters ionized gas between the Sun and the Earth, in this case,starting at a distance of about 0:5 AU from the Sun. The electrons excite waves in the gas at theplasma frequency corresponding to the electron density at that position (np ¼ 8:9 n

1=2e kHz, Eq.

E.9). The radio emission can only escape if its frequency is greater than the plasma frequency of gasthat it encounters, a situation that is possible for a density distribution that declines with distancefrom the Sun. The lower envelope of this plot denotes the plasma frequency cutoff for a decliningdensity. The initial time delay of 69 min is the time it takes for the radio emission to first reach thespacecraft which was a distance of 8.67 AU from the Sun on that date. The maximum frequenciesdetected at the earliest time are somewhat higher than shown on this plot (of order 100 MHz).(Credit: D. Gurnett, NASA/JPL, & the University of Iowa)

APPENDIX E PLASMAS, THE PLASMA FREQUENCY, AND PLASMA WAVES 375

and a group velocity (the rate at which energy, or anywavemodulation or ’information’

advances through the medium),

ng ¼ c1

vp

v

21=2ðE:8Þ

(Ref. [96]). Since the group velocity is the rate of energy transfer, it must always be less

than or equal to c, consistent with Special Relativity. The phase velocity, however, is

greater than or equal to c. This does not violate Special Relativity because no energy is

transfered at this rate. Note that, from Eqs. (E.7) and (E.8), vph vg ¼ c2.

Now, if v vp, the natural plasma oscillations are very slow in comparison to the

perturbing wave. The medium then has very little effect on the electromagnetic wave

and the wave propagates through it, as if it were transparent, with vph vg c. If

v vp (but v 6< vp), there is a strong interaction between the wave and the electrons

in the gas. In this case, vph can be much greater than c and vg can be much less. The

result is a resonance, a situation in which matter is perturbed at a rate similar to the rate

of its natural oscillation (recall resonance scattering from Appendix D.1.2). However,

ifv < vp, then there is no solution to Eqs. (E.7) and (E.8). Such a situation corresponds

to no propagation through the medium since the natural oscillations are so fast in

comparison to the disturbance produced by the wave that they essentially cancel the

disturbance. In this case, the wave reflects from the gaseous surface in much the same

way that optical light reflects from a solid5.

Expressed in cycles per second, the plasma frequency is,

np ¼ 8:9 n1=2e kHz ðE:9Þ

For a typical ISM electron density of ne ¼ 103 cm3, np ¼ 280 Hz. This is a very low

frequency in comparison to those at which astronomical measurements are made so the

ionized component of the ISM is transparent to light (note, however, that scattering and

absorption may still be occurring). However, the plasma frequency can become

important for higher density gases. For example, the reason that low frequency radio

waves do not reach the ground, as shown in the atmospheric transparency curve

of Figure 2.2, is because of reflection of these signals from the Earth’s ionosphere

(Prob. 5.1.2). Reflection of low frequency waves from plasmas is also important in

other contexts. For example, the reflection of radar signals from ionized meteor trails is

used to understand the properties of incoming meteoroids, and the reflection of man-

made Earth-based signals from the underside of the ionosphere allows us to hear radio

stations that are located over the physical horizon.

5It is interesting to note that, for similar reasons, metals reflect optical light but can become transparent at UVwavelengths.

376 APPENDIX E PLASMAS, THE PLASMA FREQUENCY, AND PLASMA WAVES

Appendix FThe Hubble Relation and theExpanding Universe

F.1 Kinematics of the Universe

As indicated in Sect. 7.2.2, the expansion of the Universe (see Figure F.1) produces the

observed Hubble Relation (Figure F.2). For galaxies that are not very distant (closest to

the origin in Figure F.2), the Hubble Relation can be written simply as the linear

function,

c z ¼ H0 d km s1 ðF:1Þ

where H0, the slope of the curve, is called the Hubble constant. With distance, d, in

Mpc,H0 is expressed in units of km s1 Mpc1 (cgs units of s1). The inverse of this is

called the Hubble time,

tH ¼ 1

H0

ðF:2Þ

which gives an approximation to the age of the Universe. For H0 ¼ 71 km s1 Mpc1

(Table G.3), tH ¼ 13:8 Gyr. A more exact age requires consideration of curvature in the

plot, although the current best value is not significantly different from tH (cf. 13.7 Gyr

determined from the fluctuations in the cosmic microwave background, see Sect. 3.1).

Although the term, c z, is often referred to as the galaxy’s recession velocity, it is more

accurately a description of the expansion of space itself. The same relation would result

even if the galaxies were fixed motionless in space (Figure F.1). Superimposed on the

recessional velocity is the peculiar velocity of each galaxy as it moves through space


under the influence of the gravitational fields of other systems near it. The peculiar

velocities result inDoppler shifts which (alongwith randomerrors) produce scatter about

the curve of Figure F.2. Peculiar velocities are small in comparison to the expansion

redshift, except for the nearest galaxies (see Example 7.3), and are always < c.

Because of the finite speed of light, galaxies that are more distant (to the right) in the

Hubble plot are seen as they were at earlier times in the history of the Universe. The

origin, with z ¼ 0, d ¼ 0, t ¼ t0 is ‘here and now’. Over cosmological scales, the

simple linear relation of Eq. (F.1) no longer applies (Prob. 7.7) and we need to allow for

curvature, meaning that we must allow for an expansion rate that changes with time. A

more general form for the Hubble Relation, then, requires aHubble parameter,HðtÞ, ofwhich theHubble constant is simply the value now (H0 ¼ Hðt0Þ), aswell as a parameter

describing the curvature called, for historical reasons, the deceleration parameter1,

qðtÞ, for which the deceleration constant is the value now (q0 ¼ qðt0Þ). These are

defined by,

HðtÞ _aðtÞaðtÞ qðtÞ €aðtÞ aðtÞ

_aðtÞ2 €aðtÞ

_aðtÞ½HðtÞ2ðF:3Þ

Figure F.1. This ‘raisin cake model’ helps to conceptualize the expansion of the Universe. As thedough (space) expands, it carries the raisins (galaxies) with it. As measured from the local raisin(the Milky Way), in time D t, the nearest raisin has moved a distance of two units (v1 ¼ 2=Dt) andthe farthest raisin has moved six units (v3 ¼ 6=Dt). Thus, the most distant raisins recede at higherspeeds than the nearby raisins, as the Hubble relation indicates. The same conclusion would bereached from any raisin so, even though the origin of the Hubble plot is at the observer, the centreof the Universe is not.

1This terminology should not be taken to imply that the universe is currently decelerating.

378 APPENDIX F THE HUBBLE RELATION AND THE EXPANDING UNIVERSE

where aðtÞ is the dimensionless scale factor introduced in Sect. 7.2.2 and we use the dot

notation to indicate differentiation with time. From the right hand side of the expres-

sion for qðtÞ, we can see that the denominator is always positive. Therefore, if qðtÞ < 0,

€aðtÞ is positive and the Universe is accelerating at time t, whereas if qðtÞ > 0 then the

Universe is decelerating at t.

To better understand the scale factor, consider the Universe to consist of concentric

shells, each of which has a radial coordinate, r, originating at the observer and ending

on a shell. As the Universe expands, r, called the comoving coordinate, is ‘attached’ to

the shell and expands with it. Since the coordinate system itself is expanding, r is a

constant with time for a given shell. The scale factor then describes the form of the

0

50000

100000

150000

200000

250000

300000

350000

2000 4000 6000 8000

Distance (Mpc)

cz (

km/s

)

Figure F.2. The Hubble Relation, shown here to z 1, using the Type Ia Supernova data of Ref.[171]. The curves plot Eq. (F.16) using H0 ¼ 71 km s1 Mpc1 (Table G.3), with the dark solid curvecorresponding to q0 ¼ 0:55 (Om ¼ 0:3, OL ¼ 0:7) and the light solid curve showing q0 ¼ 0:5(Om ¼ 1, OL ¼ 0). The observer (us) is at the origin. Scatter in the figure is due to the peculiarmotions of galaxies as well as random error in the data. At larger distances, the errors are larger.Note that the distance is the luminosity distance, defined in Eq. (F.11) and c z should only bethought of as the ‘velocity’ of expansion for z 1.

F.1 KINEMATICS OF THE UNIVERSE 379

expansion. We are now helped by the fact that our Universe appears to be flat on the

largest scales (e.g. Ref [142], Ref. [159]). This means that if two parallel beams of light

are transmitted into space, over large scales they will remain parallel (space is

Euclidean) rather than converging or diverging as they might if space had some

intrinsic curvature to it2. The proper distance, i.e. the commonly understood distance

one would obtain by laying down rulers, end to end, can then be expressed in terms of

the scale factor and comoving coordinate as,

dðtÞ ¼ aðtÞ r ðF:4Þ

We now derive the Hubble Relation for z < 1 which allows us to drop terms higher than

second order in any expansions following. Expanding the scale factor, aðtÞ in a Taylorseries (Appendix A.1) about t0 we have,

aðtÞ ¼ a0 þ _a0ðt t0Þ þ€a02ðt t0Þ2 þ . . . ðF:5Þ

where the subscript, 0, refers to values at t ¼ t0, and for simplicity, we have dropped the

functional dependence on t on the right hand side only. For example, we now write3 a0rather than aðt0Þ and the values in parentheses on the right hand side now imply

multiplication rather than a functional dependence. Using the relations of Eqs. (F.3)

evaluated now, the above becomes,

aðtÞ ¼ a0 ½1 H0ðt0 tÞ q0

2H2

0ðt0 tÞ2 ðF:6Þ

where ðt0 tÞ is called the look back time, a positive quantity since t0 > t. If we use the

definition for expansion redshift (Eq. 7.11) and a binomial expansion (Appendix A.2)

the redshift becomes,

zðtÞ ¼ H0ðt0 tÞ þ ð1þ q0

2ÞH2

0ðt0 tÞ2 ðF:7Þ

We then invert this equation to solve for tðzÞ,

ðt0 tÞ ¼ 1

H0

½z ð1þ q0

2Þ z2 ðF:8Þ

Thus, for anymeasured redshift less than 1, if we know the currently observed values of

H0 and q0, the look back time can be evaluated from Eq. (F.8).

2On small scales, the path of a photon passing by a mass will be altered (see Sect. 7.3.1), but this can be neglectedon large scales.3Since a0 is a dimensionless scale factor that refers to ‘now’, it is normally set to the value, 1. We have retained itin this development, but a0 ¼ 1 could be used throughout.


A photon, travelling radially towards us through expanding space, moves from a

larger comoving coordinate through smaller ones with increasing time. The infinite-

simal distance travelled by such a photon is determined by the Robertson–Walker

metric, an expression describing the separation between events in space–time in an

expanding homogeneous and isotropic universe. For a flat universe this distance is

c dt ¼ aðtÞ dr (cf. Eq. F.4). Then between the time the photon is emitted from the

source, t, and the time it is observed, t0, it travels from the comoving coordinate of the

source, r, to us at r ¼ 0,

Z 0

r

dr ¼ c

Z t0

t

1

aðtÞ dt ðF:9Þ

If we insert Eq. (F.6) into the right hand side of Eq. (F.9), again use a binomial

expansion, integrate, and use Eq. (F.8), we find (Prob. 7.8) the comoving coordinate of

the source as a function of its redshift,

r ¼ c z

a0 H0

½1 1

2ð1þ q0Þ z ðz < 1Þ ðF:10Þ

Although it is tempting to use Eq. (F.4) to convert the comoving coordinate, r, to a

proper distance, d, in reality, we cannot measure the proper distance. Rather we

measure the flux of an object, f , for which the intrinsic luminosity, L, is believed to

be known (see Sects. 1.1 and 1.2 for a discussion of flux and luminosity). The distance

that is inferred is the luminosity distance defined as,

dL2 L

4p fðF:11Þ

(compare Eq. F.11 to Eq. 1.9) where the observed flux is,

f ¼ L

4pða0 rÞ2ð1þ zÞ2ðF:12Þ

The term 4pða0 rÞ2 is the surface area of a sphere when and where the flux is measured

(at t ¼ t0 or ‘now’; recall that r is a constant with time for a given object). The term

ð1þ zÞ2 results from two effects. Firstly, since the wavelength is stretched by the

expansion, the received energy of a photon, Eobs, is related to the energy emitted, Eem

by (see Table I.1, Eq. 7.1),

Eobs ¼h c

l¼ h c

l0ð1þ zÞ ¼Eem

ð1þ zÞ ðF:13Þ

Thus the received flux (/ Eobs) is diminished by the same factor. Secondly, a time

interval as measured by a distant observer, Dtobs, is larger than a time interval in the

F.1 KINEMATICS OF THE UNIVERSE 381

source rest frame, Dtem, (Prob 7.9) according to,

Dtobs ¼ Dtemð1þ zÞ ðF:14Þ

Consequently, the flux (/ the number of photons received per unit time) is less again by

the same factor. It is important to note that, since the measurement of f applies to the

light when it was emitted in the past, dL as plotted in Figure F.2 therefore is the distance

to the object at the time its light was emitted and is not the distance to the object now.

Combining Eq. (F.11) and Eq. (F.12), the luminosity distance can be written4,

dL ¼ a0 rð1þ zÞ ðF:15Þ

For very small redshifts (z 1), the luminosity distance is just the proper distance of

Eq. (F.4) (measured at t ¼ t0) and is the value used in Eq. (F.1). We now need only to

substitute our expression for r from Eq. (F.10) into Eq. (F.15) (and eliminate the

diminishingly small term in z3) to arrive at the Hubble Relation for z < 1,

H0 dL ¼ c z ½1þ 1

2ð1 q0Þ z z < 1 ðF:16Þ

where we have expressed the result in a slightly rearranged form. Again, for very small

redshifts, this reduces to Eq. (F.1). For z > 1, Eq. (F.16) cannot be used since higher

order terms in the expansions become important (see Sect. F.3). Note that even though

this equation describes the relation between z and dL over a range of redshift, it is

parameterized in terms of the current values of the Hubble and deceleration parameters.

To determine (calibrate) this relation, it is straightforward to measure z from the

wavelength shift of spectral lines in distant sources. However, measurements of

distance are more challenging. Those objects for which the absolute luminosity is

believed to be known are called standard candles, as mentioned in Sect. 5.4.2. The flux

is measured and Eq. (F.11) is then used to find dL. A variety of standard candles are

used, but for the larger scales, these candles must be quite bright in order to be seen

over cosmological scales. The Cepheid variables discussed in Sect. 5.4.2 are only

bright enough to measure distances to the closer galaxies. Fortunately, good data come

from Type Ia supernovae (SNe, as plotted in Figure F.2) which are believed to result

from the thermonuclear explosion of a white dwarf star5 when it accretes more gas

from a nearby companion than its internal structure can support. As the explosion

proceeds, we observe a light curve which is a plot of increasing, then decreasing

brightness with time. From measurements of Type Ia SNe of known distance, it has

4Another common measure of distance in astronomy is called the angular size distance defined as das ¼ x=u,where x is the physical linear diameter of the source in the plane of the sky and u is its apparent angular diameteras seen from the Earth. It is used when angular sizes are measured rather than luminosities and is related to theluminosity distance via dL ¼ dasð1þ zÞ2.5White dwarfs are very compact stars having an upper mass limit of 1:4 M and a size approximately that of theEarth. They are the end point of evolution of a lowmass star like the Sun once the outer layers have been dispersed(Sect. 3.3.2). This type of supernova therefore has a different origin than the Type II supernovae discussed in Sect.3.3.3.


been found that the peak luminosity of this curve is almost constant for all Type Ia SNe.

Moreover, those SNe with higher peak luminosities have a slower rate of decline.

Therefore, a comparison of the rate of decline of a distant supernova with supernovae

of known luminosity allows one to determine the intrinsic peak luminosity of the

distant supernova to high accuracy. Once the intrinsic luminosity is known, the distance

can be determined as described above. The Hubble relation can then be plotted from the

redshifts and distances of a sufficient number of these standard candles.H0 and q0 may

then be found (Prob. 7.10) via fits to Eq. (F.16). With H0 and q0 known, the Hubble

relation has then been calibrated.

Supernovae have been observed in only a small fraction of all galaxies and other

standard candles are faint and difficult to observe in the more distant systems. This is

why the calibrated Hubble relation is a very important tool for determining distances to

galaxies. One need only measure a redshift and a distance is found through this relation

(e.g. Eq. F.16 for z < 1Þ. Because calibration is difficult, however, the value of the

Hubble constant has been the subject of much debate over the years, and the value of the

deceleration constant even more so since it characterizes a change in H with time. It is

currently believed that H0 is known to within about 10 per cent (Ref. [39]). Since q0measures curvature, its value is revealed only in the most distant objects which are also

subject to themost error. Figure F.2 shows a clear trendwhich suggests that our Universe

is currently accelerating with time since q0 < 0 (see also next section). There has been

some suggestion that systematic errors may be creating this trend. For example, if the

supernovae were intrinsically fainter in the distant past, then they would appear farther

away, causing the curve to turn over as observed. Another possibility is that some

intervening ‘grey dust’ (dust whose effects on the signal are independent of wavelength

and therefore would be difficult to detect) would make them appear fainter. However,

independent data from spatial fluctuations in the cosmic background radiation (see

Figure 3.2, Sect. 3.1) are consistent with the requirement of an acceleration term (Ref.

[159]). This has been one of the greatest surprises of modern cosmology.

F.2 Dynamics of the Universe

What is producing this large scale behaviour of the Universe?We have dealt, so far, only

with the kinematics of the expanding Universe.We have not yet considered the dynamics

which would relate this expansion to the forces that drive it. For this, Einstein’s Field

Equations of General Relativity are used together with the Robertson–Walker metric,

resulting in an expression called the Friedmann Equation. In the version used here, we

adopt a flat universe, neglect radiation pressure which is important only in the radiation-

dominated era of the early universe at z 1000, and modify the result to include an

acceleration term expressed as the cosmological constant, L (cgs units of s2). The

result is,

_a

a

2 8pG r

3 L

3¼ 0 ðF:17Þ

F.2 DYNAMICS OF THE UNIVERSE 383

where r (g cm3) is the total mass density of the Universe including all matter, both

light and dark6. It is more general to express r andL in terms of energy densities, e (ergcm3). The mass density can be expressed in terms of the energy density of matter and

the cosmological constant can be thought of in terms of the energy density of a vacuum.

For matter, we use Einstein’s famous equation for the equivalence of mass and energy,

expressed as a density,

eM ¼ r c2 ðF:18Þ

and for the accleration term, we define an energy density,

eL L c2

8pGðF:19Þ

With these relations, the Friedmann Equation is rewritten using Eq. (F.3),

H2 8pG

3 c2ðeM þ eLÞ ¼ 0 ðF:20Þ

We now define a critical energy density,

ec 3H2 c2

8pGðF:21Þ

as well as density parameters,

OM eM

ec¼ 8pG r

3H2OL e

ec¼ L

3H2ðF:22Þ

where the right hand side of each equation makes use of Eq. (F.19) and Eq. (F.21). So

the Friedmann Equation reduces, for a flat universe, to the simple expression,

OM þ OL ¼ 1 ðF:23Þ

Note that both OM and OL change with time, but their sum should always equal 1 in a

flat universe with an acceleration term. If there were no cosmological acceleration

term, then OM ¼ 1 ) eM ¼ ec. Thus, the critical energy density is the energy density

associated with matter in a flat universe, in the absence of a value of OL. In a universe

with an acceleration term, bothOM andOL must be< 1. BecauseOM includes all of the

contents of the universe, one could also break it down into its various components. For

example, dark matter, OD ¼ eD=ec, and matter that radiates, OL ¼ eL=ec, so that

OD þ OL ¼ OM. Dark matter could be expressed as baryonic and non-baryonic

(OB þ ON ¼ OD), and so on. Determining these quantities is an important goal of

6It also includes the radiation content, although this is negligible for z < 1000 and so will be neglectedthroughout.


modern cosmology since they tell us about the kind of universewe live in. Thus we now

need to relate the density parameters of Eq. (F.22) to observable quantities, such as H0

and q0, which can be measured from the Hubble Relation.

To do this, we make use of conservation of energy, i.e. the total energy content of the

Universe associated with matter (which includes radiation), e, within any comoving

radius, r, must not change with time. Expressing this as e ¼ rðr aÞ3 c2 ¼ constant

(having used Eq. F.4) and differentiating with respect to time leads to

ð _r a3 þ 3 r a2 _aÞ ¼ 0. Solving for _r and using Eq. (F.3),

_r ¼ 3 rH ðF:24Þ

Now differentiating Eq. (F.17) with respect to time, and substituting Eq. (F.17) and Eq.

(F.24) into the result yields,

€a

aþ 4pG r

3 L

3¼ 0 ðF:25Þ

This is the acceleration equation, sometimes called the Second Friedmann Equation.

Using Eq. (F.3), Eq. (F.22), and Eq. (F.21), this becomes,

q ¼ OM

2 OL ðF:26Þ

If there were no cosmological constant (OL ¼ 0), then a flat universe would require

OM ¼ 1 and q ¼ 0:5.The above equations apply to any time, t, and therefore they also apply now, so any

quantity that has a dependence on t can be expressed with its present day value. Thus,

Eq. (F.22), Eq. (F.23) and Eq. (F.26) become,

OM0 ¼8pG r03H0

2; OL0 ¼

L

3H02; OM0 þ OL0 ¼ 1; q0 ¼

OM0

2 OL0 ðF:27Þ

Often, the subscript, 0, is dropped and the values are taken to be understood as the

respective values now. These are extremely useful relations since they relate the

observed values, q0, and H0, obtainable from the Hubble Relation, to quantities that

describe the fundamental nature of our Universe. Notice that Eqs. (F.27) give us four

equations in the four unknowns, OM0, OL0, r0, and L. Thus, all quantities can, in

principle, be determined. By considering how these quantities vary with time, more-

over, the history and nature of the expansion may be revealed. For example, dividing

the Friedmann equation by H02, using r ¼ r0ða0=aÞ

3(since total mass is conserved)

and applying our previous definitions evaluated today, we find,

H2

H02¼ OM0ð

a0

aÞ3 þ OL0 ðF:28Þ

F.2 DYNAMICS OF THE UNIVERSE 385

The only time-variable quantities in this equation are H and a. This equation makes it

quite clear that the expansion of the universe will becomemore andmore dominated by

the acceleration term in later times as a increases (Prob. 7.11), if this is indeed the kind

of universe we live in.

The Hubble Relation shown in Figure F.2 shows two curves, one with OL0 ¼ 0,

OM0 ¼ 1, q0 ¼ 0:5 (light curve) and one with OL0 ¼ 0:7, OM0 ¼ 0:3, q0 ¼ 0:55(dark curve). The latter set of values are a better match to the data. As pointed out in

Sect. F.1, q0 < 0 implies that the Universe is currently accelerating. From Eq. (F.22)

and H0 from Table G.3, the latter set of values gives r0 ¼ 2:8 1030 g cm3 and

L ¼ 1:1 1035 s2. It is these quantities which have been presented as percentages

in Figure 3.4, but the Figure 3.4 values are taken from a broader range of measure-

ments than outlined in this Appendix. In that figure, ‘dark energy’ (not to be confused

with dark matter) refers to whatever energy source is producing OL0 and ‘matter’

refers to all matter, both light and dark, that is represented by OM0 (see also the

discussion in Sect. 3.2). It is clear that, provided there are no systematic errors in

Figure F.2, a matter dominated universe is unable to describe the universe we live in.

Since an acceleration term is needed, there must be some kind of dark energy that is

producing it. The form that this dark energy might take is currently unknown.

Theoretical cosmologists are actively pursuing a variety of possibilities including

the concept of quintessence which allows the acceleration term to vary with time,

rather than being expressed in the form of a cosmological constant. It may be that

advances in the realm of fundamental physics will be required before its nature will be

fully revealed.

It is astonishing that more than half of the energy of the universe appears to be

in a mysterious form that accelerates it and, of the remainder, most of the matter is

dark (Sect. 3.2). Thus, our knowledge of the universe is gleaned from only a tiny

fraction of its contents. Remarkably, it indeed appears to be possible to discern

the large-scale structure and behaviour of the Universe from the visible tracers

within it.

F.3 Kinematics, dynamics and high redshifts

Since we now have a clearer view as to the kind of Universe we live in, it is becoming

more common to write the kinematic equations of Sect. F.1 in terms of the dynamical

expressions of Sect. F.2. For example, substituting the expression for q0 as well as

OL0 þ OM0 ¼ 1 from Eqs. (F.27) into Eq. (F.10), gives a dynamical expression for the

comoving distance when z < 1,

r ¼ c z

a0 H0

1 3

4OM0 z

ðz < 1Þ ðF:29Þ


Using Eq. (F.15) and eliminating terms in z3 which are small for z < 1, this can be

converted to a luminosity distance,

dL ¼ c z

H0

1þ

1 3

4OM0

z

ðz < 1Þ ðF:30Þ

Eq. (F.30) is another way, besides the kinematical result of Eq. (F.16), of relating the

redshift, z, to the distance, dL, of the Hubble Relation (Figure F.2).

In Sect. F.1, we did not consider any redshifts higher than 1. Currently, the highest

measured redshifts are of order z 7 (Ref. [89]) and there is every reason to believe

that higher redshifts will be routinely measured as more powerful telescopes are trained

towards the faintest objects in the deepest reaches of the Universe. For z > 1, however,

the higher order terms of earlier expansions cannot be neglected and Eqs. (F.16) and

(F.30) can no longer be used to describe the Hubble Relation or to determine the

distance. For the sake of completeness, we provide here the full expression for

comoving distance for a flat Universe with a cosmological constant in which

OM þ OL ¼ 1,

r ¼ c

a0 H0

Z z

0

dzOM0 z

3 þ 3OM0 z2 þ 3OM0 zþ 1

1=2 ðF:31Þ

As before, this expression can be used with Eq. (F.15) to determine a luminosity

distance, or any other distance (see Footnote 4 of this Appendix) that may be desired.

F.3 KINEMATICS, DYNAMICS AND HIGH REDSHIFTS 387

Appendix GTables and Figures

Table G.1. The Greek alphabet

Capital Small Name

A a alpha

B b beta

G g gamma

D d delta

E e epsilon

Z z zeta

H h eta

Q u theta

I i iota

K k kappa

L l lambda

M m mu

N n nu

J j xi

O o omicron

P p pi

P r rho

S s sigma

T t tau

Y y upsilon

F f phi

X x chi

C c psi

V v omega

Astrophysics: Decoding the Cosmos Judith A. Irwin# 2007 John Wiley & Sons, Inc. ISBN: 978 0-470-01305-2(HB) 978-0-470-01306-9(PB)

Table G.2. Fundamental physical dataa

Symbol Meaning Value

Physical constants

c Speed of light in vacuum 2:997 924 58 1010 cm s1

G Universal gravitational constant 6:6742ð10Þ 108 cm3 g1 s2

g Standard gravitational acceleration (Earth) 9:806 65 102 cm s2

k Boltzmann constant 1:380 6505ð24Þ 1016 erg K1

8:617 343ð15Þ 105 eV K1

h Planck constant 6:626 0693ð11Þ 1027 erg s

N bA Avogadro’s constant 6:022 1415ð10Þ 1023 mol1

R Molar gas constant 8:314 472ð15Þ 107 erg mol1 K1

R1 Rydberg constant 109 737:315 68 525ð73Þ cm1

Atomic and nuclear data

ec Atomic unit of charge 4:803 250ð21Þ 1010 esu

eV Electron volt 1:602 176 53ð14Þ 1012 erg

me Electron mass 9:109 3826ð16Þ 1028 g

mn Neutron mass 1:674 927 28ð29Þ 1024 g

mp Proton mass 1:672 621 71ð29Þ 1024 g

uamu Unified atomic mass unit 1:660 538 86ð28Þ 1024 g

r de Classical electron radius 2:817 940 325ð28Þ 1013 cm

lC Compton wavelength 2:426 310 238ð16Þ 1010 cm

lCnNeutron Compton wavelength 1:319 590 9067ð88Þ 1013 cm

lCpProton Compton wavelength 1:321 409 8555ð88Þ 1013 cm

a0 Bohr radius 0:529 177 2108ð18Þ 108 cm

sT Thomson cross section 0:665 245 873ð13Þ 1024 cm2

Radiation constants

s Stefan–Boltzmann constant 5:670 400ð40Þ 105 erg s1 cm2 K4

b Wien’s Displacement Law constant 2:897 7685ð51Þ 101 cm K

ae Radiation constant 7:565 91ð25Þ 1015 erg cm3 K4

aData have been taken from Ref. [168], converted to cgs units, unless otherwise indicated. The values inparentheses, where provided, represent the standard error of the last digits which are not in parentheses.bThis is the number of particles in one mole (mol) of material.cFrom Ref. [96]. An esu is an ‘electrostatic unit’. The corresponding SI unit of charge is 1:602 1019 C. Notethat the force equation in cgs units, F ¼ ðq1 q2Þ=r2, does not have a constant of proportionality for F in dynes,q1; q2 in esu, and r in cm.dre ¼ e2=ðme c

2Þ.e Ref. [44].

390 APPENDIX G TABLES AND FIGURES

Table G.3. Astronomical dataa

Symbol Meaning Value

Distance

AU Astronomical unit 1.495 978 706(2)1013 cm

pc Parsec 3.085 677 582 1018 cm

ly Light year 9.460 730 47 1017 cm

Time

yr Tropical yearb 3.155 692 58 107 s

yr Sidereal yearc 3.155 814 98 107 s

day Mean sidereal dayc 23h 56m 04.090 524s

Earth

M Earth mass 5.973 70(76) 1027 g

R Mean Earth radiusd 6.371 000 108 cm

Requ Earth equatorial radiusd 6.378 136 108 cm

Rpol Earth polar radiusd 6.356 753 108 cm

r Mean Earth densityd 5.515 g cm3

Moon

Mm Moon massd 7.353 1025 g

Rm Mean Moon radiusd 1.738 2 108 cm

rm Mean Moon densityd 3.341 g cm3

rm Mean Earth – Moon distanced 3.844 01(1) 1010 cm

Sun

M Solar mass 1.989 1(4) 1033 g

R Solar radius 6.955 1010 cm

r Mean solar density 1.41 g cm3

L Solar luminosityd 3.845(8) 1033 erg s1

Teff Solar effective temperaturee 5781 K

S Solar constantf 1.367 106 erg s1 cm2

mV Solar apparent visual magnitudee 26.76

MV Solar absolute visual magnitudee 4.81

Mb Solar absolute bolometric magnitudee 4.74

u Angular diameter of Sun from Earth 32.00

Galaxy

R Sun Galactocentric distanceg 8.5(5) kpc

VLSR Velocity of the Local Standard of Resth 220(20) km s1

Extragalactic

H0 Hubble constanti 71þ43 km s1 Mpc1

TCMB Cosmic Microwave Background temperature 2.726 0.005 K

aRef. [96] unless otherwise indicated. Values in parentheses indicate the one standard deviation uncertainties inthe last digits.bEquinox to equinox.cFixed star to fixed star.dRef. [44].eRef. [18].f Flux of the Sun at distance of the Earth.gDistance of Sun from the centre of the Galaxy. A value of 8 kpc is often used (e.g. Sect. 10.2.1).hVelocity of an object at a Galactocentric distance of R in circular orbit about the centre of the Galaxy. TheSun’s velocity about the Galactic centre agrees with this value to within the quoted error.iRate of expansion of the Universe. The value is from Ref. [159]; errors are likely closer to 10 per cent (Ref.[39]).

Table

G.4.

Planetarydataa

Equat

ori

alR

ota

tion

dia

met

erM

ass

Den

sity

per

iodb

Alb

edod

ae

Pla

net

(km

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bla

tenes

s(E

arth

=1)

(gcm

3)

(day

s)In

cl.c

(Bond)

(AU

)ef

Mer

cury

4879

00:0

55

274

5:4

358:6

46

0:0

0:1

19

0:3

871

0:2

056

Ven

us

12

104

00:8

15

005

5:2

4243:0

19

2:6

0:7

50

0:7

233

0:0

068

Ear

th12

756

1=298

1:0

00

000

5:5

20:9

973

23:4

0:3

06

1:0

000

0:0

167

Mar

s6

792

1=148

0:1

07

447

3:9

41:0

260

25:2

0:2

50

1:5

237

0:0

935

Jupit

er142

980g

1=15:4

317:8

33

1:3

30:4

101h

3:1

0:3

43

5:2

020

0:0

490

Sat

urn

120

540g

1=10:2

95:1

63

0:6

90:4

440

26:7

0:3

42

9:5

752

0:0

568

Ura

nus

51

120g

1=43:6

14:5

36

1:2

70:7

183

82:2

0:3

00

19:1

315

0:0

501

Nep

tune

49

530g

1=58:5

17:1

49

1:6

40:6

712

28:3

0:2

90

29:9

681

0:0

086

Plu

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2390

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0:0

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57:4

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Table G.5. Data for human visiona

Quantity Value

Pupil diameter (dark-adapted) 0.8 cm

(daylight) 0.2 cm

Approx. number of photoreceptor cellsb

Rods 108

Cones 6:5 106

Retina effective area 10 cm2

Focal length 2.5 cm

Resolution of human eye 10c

Wavelength response of human eye:

Bright-adapted (photopic) visible range 400–700 nm

Dark-adapted (scotopic) visible range 400–620 nm

Wavelength of peak sensitivity of eye:

Bright-adapted 555 nm

Dark-adapted 507 nm

Representative wavelengths of colours:

Violet 420 nm

Blue 470 nm

Green 530 nm

Yellow 580 nm

Orange 610 nm

Red 660 nm

aWavelengths are from Ref. [71]. Note that values for the eye are typical,but vary with individual.bIn order to register as a separate source, each photoreceptor cell must beseparated by at least one other unexcited photoreceptor cell. Dark-adaptedvision response is primarily due to rods.cThis value varies depending on the individual and lighting conditions.

Table G.6. The electromagnetic spectruma

Wavelength range Frequency range Energy rangeb

Waveband (cm) (Hz) (eV)

Radioc 1 31010 1.2104

(Microwave)c (100 ! 0.1) (3108 ! 31011) (1.2106 ! 1.2103)

Millimetre –

Submillimetrec 1 ! 0.01 31010 ! 31012 1.2104 ! 1.2102

Infrared 0.01 ! 104 31012 ! 31014 1.2102 ! 1.2

Optical 104 ! 3105 31014 ! 1015 1.2 ! 4.1

Ultraviolet 3105 ! 106 1015 ! 31016 4.1! 124

X-ray 106 ! 109 31016 ! 31019 124 ! 1.2105

Gamma-ray 109 31019 1.2105

aThere is some variation as to where the ‘boundaries’ of the various wavebands lie.bThe energy, in electron volts, can be computed from the definition of an electron volt (Table G.2) and therelation between energy and wavelength or frquency (Table 1.1), hence 1 eV => l ¼ 1:24 104 cm andn ¼ 2:42 1014 Hz.cIn astrophysics, the radio band is taken to include microwave frequencies and occasionally to include themillimetre – submillimetre bands as well.

APPENDIX G TABLES AND FIGURES 393

Table G.7. Stellar dataa, Luminosity Class V (Dwarfs)

Spectral type B–V V–R MV BC Teff(K) R=R M=M

O3 4.3? 48000 50?

O5 5? 4.3? 44000 30?

O6 0.32 0.15 4.8 4.25 43000 12? 25?

O8 0.31 0.15 4.1 3.93 37000 10 20?

B0 0.29 0.13 3.3 3.34 31000 7.2 17

B1 0.26 0.11 2.9 2.6 24100 5.3 10.7

B2 0.24 0.1 2.5 2.2 21080 4.7 8.3

B3 0.21 0.08 2 1.69 18000 3.5 6.3

B4 0.18 0.07 1.5 1.29 15870 3 5

B5 0.16 0.06 1.1 1.08 14720 2.9 4.3

B8 0.1 0.02 0 0.6 11950 2.3 3

A0 0 0.02 0.7 0.14 9572 1.8 2.34

A2 0.06 0.08 1.3 0 8985 1.75 2.21

A5 0.14 0.16 1.9 0.02 8306 1.69 2.04

A7 0.19 0.19 2.3 0.02 7935 1.68 1.93

F0 0.31 0.3 2.7 0.04 7178 1.62 1.66

F2 0.36 0.35 3 0.04 6909 1.48 1.56

F5 0.44 0.4 3.5 0.02 6528 1.4 1.41

F8 0.53 0.47 4 0.01 6160 1.2 1.25

G0 0.59 0.5 4.4 0.05 5943 1.12 1.16

G2 0.63 0.53 4.7 0.08 5811 1.08 1.11

G5 0.68 0.54 5.1 0.11 5657 0.95 1.05

G8 0.74 0.58 5.6 0.16 5486 0.91 0.97

K0 0.82 0.64 6 0.23 5282 0.83 0.9

K2 0.92 0.74 6.5 0.3 5055 0.75 0.81

K3 0.96 0.81 6.8 0.33 4973 0.73 0.79

K5 1.15 0.99 7.5 0.43 4623 0.64 0.65

K7 1.3 1.15 8 0.54 4380 0.54 0.54

M0 1.41 1.28 8.8 0.72 4212 0.48 0.46

M2 1.5 1.5 9.8 0.99 4076 0.43 0.4

M5 1.6 1.8 12 1.52 3923 0.38 0.34

aRef. [68]. These values should be considered typical, but individual stars may differ from the values quotedhere. Blanks imply that the data are not sufficiently well known to be quoted here.


Table G.8. Stellar dataa, Luminosity Class III (Giants)

Spectral Teff

type B–V V–R MV BC (K) R=R

F0 0.31 1 0.04 7178 7

F2 0.36 0.9 0.04 6909

F5 0.44 0.8 0.02 6528

F8 0.54 0.7 0.02 6160 8

G0 0.64 0.6 0.09 5943 9

G2 0.76 0.5 0.17 5811 10

G5 0.9 0.69 0.4 0.29 5657 11

G8 0.96 0.7 0.3 0.33 5486 12

K0 1.03 0.77 0.2 0.37 5282 14

K2 1.18 0.84 0.1 0.45 5055 17

K3 1.29 0.96 0.1 0.53 4973 21

K5 1.44 1.2 0 0.81 4623 40

K7 1.53 0.1 1.15 4380 60

M0 1.57 1.23 0.2 1.36 4212 100

M2 1.6 1.34 0.2 1.52 4076 130

M5 1.58 2.18 0.2 3923

aRef. [68]. See notes to Table G.7. Masses of Type III giants are not well known but are likely 2–2:5 M.


Table G.9. Stellar dataa, Luminosity Class Ib (Supergiants)

Spectral Teff

type B–V V–R MV BC (K) R=R

A0 0.01 0.03 5 0.12 9550 70

A2 0.05 0.07 5 0.02 9000

A5 0.1 0.12 5 0.01 8500

A7 0.13 4.9 0.02 8300

F0 0.16 0.21 4.8 0.02 8030 80

F2 0.21 0.26 4.8 0.02 7780

F5 0.33 0.35 4.7 0.04 7020

F8 0.55 0.45 4.6 0.02 6080

G0 0.76 0.51 4.6 0.18 5450 100

G2 0.87 0.58 4.6 0.26 5080

G5 1 0.67 4.5 0.35 4850

G8 1.13 0.69 4.5 0.41 4700

K0 1.2 0.76 4.5 0.47 4500 110

K2 1.29 0.85 4.5 0.53 4400

K3 1.38 0.94 4.5 0.68 4230 130

K5 1.6 1.2 4.5 1.52 3900 200

K7 1.62 4.5 3870

M0 1.65 1.23 3850 230

M2 1.65 1.34 3800

aRef. [68]. See notes to Table G.7. Masses of Type Ib supergiants are not well known but are likely 10 M.


Table G.10. Solar interior modela

Ld Te rf

R=Rb M=M

c(erg s1) (K) (g cm3)

1.14E-03 1.64E-07 5.66E+27 1.58E+07 1.56E+02

2.48E-02 1.64E-03 5.50E+31 1.56E+07 1.49E+02

1.03E-01 8.29E-02 1.84E+33 1.30E+07 8.57E+01

1.87E-01 3.00E-01 3.52E+33 9.82E+06 3.95E+01

2.73E-01 5.44E-01 3.83E+33 7.40E+06 1.62E+01

3.55E-01 7.21E-01 3.85E+33 5.80E+06 6.50E+00

4.39E-01 8.36E-01 3.85E+33 4.64E+06 2.56E+00

5.29E-01 9.09E-01 3.85E+33 3.72E+06 1.01E+00

6.25E-01 9.53E-01 3.85E+33 2.94E+06 4.02E-01

7.22E-01 9.77E-01 3.85E+33 2.09E+06 1.76E-01

8.04E-01 9.90E-01 3.85E+33 1.33E+06 8.80E-02

8.66E-01 9.96E-01 3.85E+33 8.41E+05 4.41E-02

9.09E-01 9.99E-01 3.85E+33 5.34E+05 2.21E-02

9.39E-01 1.00E+00 3.85E+33 3.41E+05 1.12E-02

9.59E-01 1.00E+00 3.85E+33 2.19E+05 5.58E-03

9.72E-01 1.00E+00 3.85E+33 1.43E+05 2.76E-03

9.81E-01 1.00E+00 3.85E+33 9.57E+04 1.34E-03

9.86E-01 1.00E+00 3.85E+33 6.44E+04 6.52E-04

9.90E-01 1.00E+00 3.85E+33 4.48E+04 3.10E-04

9.93E-01 1.00E+00 3.85E+33 3.35E+04 1.39E-04

9.95E-01 1.00E+00 3.85E+33 2.65E+04 5.93E-05

9.96E-01 1.00E+00 3.85E+33 2.18E+04 2.43E-05

9.97E-01 1.00E+00 3.85E+33 1.84E+04 9.69E-06

9.98E-01 1.00E+00 3.85E+33 1.58E+04 3.81E-06

9.988E-01 1.00E+00 3.85E+33 1.36E+04 1.50E-06

9.993E-01 1.00E+00 3.85E+33 1.16E+04 6.04E-07

9.998E-01 1.00E+00 3.85E+33 8.72E+03 2.76E-07

1.0000E+00 1.00E+00 3.85E+33 5.67E+03 1.55E-07

1.0001E+00 1.00E+00 3.85E+33 5.11E+03 9.61E-08

1.0002E+00 1.00E+00 3.85E+33 4.82E+03 5.62E-08

1.0003E+00 1.00E+00 3.85E+33 4.70E+03 3.10E-08

1.0004E+00 1.00E+00 3.85E+33 4.65E+03 1.68E-08

1.0005E+00 1.00E+00 3.85E+33 4.64E+03 9.07E-09

1.0006E+00 1.00E+00 3.85E+33 4.63E+03 4.87E-09

1.0007E+00 1.00E+00 3.85E+33 4.63E+03 2.58E-09

1.0008E+00 1.00E+00 3.85E+33 4.63E+03 1.34E-09

1.0009E+00 1.00E+00 3.85E+33 4.63E+03 6.67E-10

1.0011E+00 1.00E+00 3.85E+33 4.63E+03 3.03E-10

aModelled conditions for the interior of the Sun adopting Teff ¼ 5779:6 K, L ¼ 3:851 1033 erg s1,X¼ 0:708, Z¼ 0:02R ¼ 6:9598 1010 cm and M ¼ 1:98910 1033g. (Ref. [70]). (Reproduced bypermission of D. Guerther)bRadius as a fraction of the Solar radius.cMass as a fraction of the Solar mass.dLuminosity.eTemperature.f Mass density.


Table G.11. Solar photosphere modela

logðt0Þb Tc logðPgÞd logðPeÞe logðk0=PeÞ f xg

(K) [log(dyn cm2)] [log(dyn cm2)] [log(cm3 s2 g2)] (km)

4.0 4310 3.13 1.42 1.22 476

3.8 4325 3.29 1.28 1.23 443

3.6 4345 3.42 1.15 1.24 415

3.4 4370 3.54 1.03 1.25 389

3.2 4405 3.66 0.92 1.27 365

3.0 4445 3.78 0.80 1.28 340

2.8 4488 3.89 0.69 1.30 317

2.6 4524 4.00 0.58 1.32 293

2.4 4561 4.11 0.48 1.33 269

2.2 4608 4.22 0.37 1.35 245

2.0 4660 4.33 0.26 1.37 221

1.8 4720 4.44 0.14 1.40 196

1.6 4800 4.54 0.01 1.42 172

1.4 4878 4.65 0.11 1.45 147

1.2 4995 4.76 0.26 1.49 122

1.0 5132 4.86 0.42 1.54 96

0.8 5294 4.96 0.60 1.58 73

0.6 5490 5.04 0.83 1.64 51

0.4 5733 5.12 1.10 1.71 31

0.2 6043 5.18 1.43 1.79 14

0.0 6429 5.22 1.80 1.88 0

0.2 6904 5.26 2.21 1.99 11

0.4 7467 5.28 2.64 2.10 19

0.6 7962 5.30 2.96 2.17 26

0.8 8358 5.32 3.20 2.22 33

1.0 8630 5.43 3.34 2.25 41

1.2 8811 5.36 3.46 2.26 50

aModelled conditions for the surface region of the Sun (Ref. [68]). The model assumes that the solar effectivetemperature is Teff ¼ 5770 K and that logðgÞ ¼ 4:44, where g (cm s2) is the surface gravity.bLog of the optical depth (Sect. 5.4.1) at the reference wavelength of 500 nm. Logðt0Þ ¼ 0 represents the‘surface’ of the Sun’s photosphere.cTemperature at the corresponding optical depth.dLog of the gas pressure.eLog of the electron pressure.f Log of the ratio of the mass absorption coefficient (Sect. 5.4.1), k0 (cm2 g1) at the reference wavelength of 500nm, to Pe.gGeometrical depth, where 0 corresponds to the photosphere surface and positive values are below the surface.


Table G.12. Acronym key to bibliography

A&A Astronomy & Astrophysics

A&ARv Astronomy & Astrophysics Review

A&AS Astronomy and Astrophysics Supplement Series

AIP Conf. Proc American Institute of Physics Conference Proceedings

AJ The Astronomical Journal

AO Applied Optics

ApJ The Astrophysical Journal

ApJS Astrophysical Journal Supplement Series

ARAA Annual Review of Astronomy and Astrophysics

AREPS Annual Review of Earth and Planetary Sciences

ASP Conf. Ser. Astronomical Society of the Pacific Conference Series

Astron. Nachr. Astronomische Nachrichten

GeoRL Geophysical Research Letters

JApA Journal of Astrophysics and Astronomy

JGR Journal of Geophysical Research

JKAS Journal of the Korean Astronomical Society

JOSA Journal of the Optical Society of America

JQSRT Journal of Quantitative Spectroscopy & Radiative Transfer

MNRAS Monthly Notices of the Royal Astronomical Society

Natur Nature

NewA New Astronomy

NewAR New Astronomy Reviews

OE Optical Engineering

PASJ Publications of the Astronomical Society of Japan

PASP Publications of the Astronomical Society of the Pacific

Phys. Rev. Physical Review

Rev. Mod. Phys Reviews of Modern Physics

Sci Science

SIAMR Society of Industrial and Applied Mathematics Review


Main Sequence Calibration

.5e4

.1e5

.2e5

T_eff

–0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

B-V

Figure G.1. B–V colour – temperature calibration for the Main Sequence, from the data of Table G.7

Figure G.2. Solar interior model from the data of Table G.10. The temperature is in kelvins anddensity in g cm3.


References

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3. Afonso, C., et al., 2003, A&A, 400, 951

4. Alcock, C., et al., 2001, ApJ, 550, L169

5. Alcock, C., et al., 1998, ApJ, 499, L9

6. Allen, G. E., et al., 1997, ApJ, 487, L97

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8. Anderson, M., & Rudnick, L., 1996, ApJ, 456, 234

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Index

Items in italic refer to figures, while items appearing in bold refer to tables.

Astrophysical flux 8

Aberration 34

chromatic

Absorption 153, 165, 168, 172

coefficient 175

free-free 169

geometry – 154

Abundances 70, 82

elements 78

primordial 70

solar 78

Active galactic nucleus (AGN) 157, 172,

181, 290

Cen A – 8

NGC 1068 158

Adaptive optics 45, 48

diagram 46

improvement 47

Airy disk 39

Albedo 134

of dust 370

of Earth 135

Alfven waves 262

Alpha particles 112

Annihilation fountain 327

Astronomical data – 392

Atmosphere 41

cosmic 174

spectral response 30

reddening in–see reddening atmospheric

refraction in–see Refraction atmospheric

transparency curve 31

Atmospheric cell 43–44, 48

Aurora

australis 262

borealis 262

Balmer series, see Hydrogen, Balmer

series line ratios 305

data 349

Baryonic matter 68–69

Beam

angleþ solid angle 54–55

switching 41

Bending angle 220

Bias correction 49

Big Bang 65

Binding energy 72

Black body radiation 123–134

Black holes 77, 220

supermassive 181, 327

Blueshift 210


Bohr magneton 353

Bohr model 347

Bolometric

correction 23

quantities 4

Boltzmann

Equation 97

factor 86, 147

Bosons 146

Bound-bound processes 177, 280

Bound-free processes 177

see also ionization

Bremsstrahlung radiation 237

determining source properties 242

geometry 237

H II regions 242

magneto 255

thermal, Spectrum 241

X-ray-emitting gas 246

Brightness 12

see also intensity

Broad-band emission 236

Brown dwarf 71, 335

Calibration 49, 50

absolute 50

Case A recombination 304

Case B recombination 304

Case studies 325

Central engine 327

see also AGN

Cepheids 179, 382

Chandrasekhar limit 74

Charge coupled device (CCD) 19, 36–37

Classification, stellar (see spectral types)

Closed box model 81

Cluster fitting 333

CO molecule 281, 311–313

in M51 313

relation to H2 312

Collisions

damping constant 295

elastic 84, 89

mean time between 90

rate coefficient 91

rate 90–91

sample parameters 92

Colour index 23, 131

Colour-magnitude diagram (CMD) 23

NGC 6397 332

see also Hertzsprung – Russell diagram

Solar neighbourhood 24

Colour-temperature calibration 391

Column density 308, 310, 312

Comoving coordinate 379

Complex

catastrophe 275

scattering (see Scattering, Compton)

spectra 253, 319–325

wavelength 368

Conduction

electrical 188

thermal 188

Continuum

characteristics of 236

emission 5, 234, 235

Convection 188

Convolution 40, 293, 340

Cooling curve, white dwarf 333

Cosmic microwave background (CMB) 65,

125, 275

fluctuations 67

Cosmic rays xvii, 112

composition 112

origin of 117

particle spectrum 115

power law energy spectrum 114, 265

showers 112

spectrum 113

synchrotron radiation 255

tracks xviii

Cosmic web 161

Simulation 162

Cosmological constant 383

Cosmological parameters 69, 276,

384, 386

Covering factor 15

Critical frequency 264

Critical gas density 289

Sample values 290

Cross section

absorption 154, 168

effective 91, 175

Klein-Nishina 368

408 INDEX

K-N 369

Sample photon 160

scattering 154, 155

Cyclotron radiation 255, 258, 262

geometry 256

Cygnus star forming complex 329, 331

Dark ages 16, 170

Dark correction 49

Dark energy 69, 386

Dark matter xx, 66, 384

Data cube 55

illustration 56

Deceleration constant 378

Deceleration parameter 378

Degeneracy 352

electron 73, 74, 89

neutron 77

Diffraction pattern 39

Diffuse ionized gas (DIG)

245, 299

Diffusion

radiative 95–96, 188

Dimensional analysis xxv

Disks

accretion 213

dusty 141

geometry of 214

rotating 213

Dissociation 169

Distance

angular size 382

comoving 379

comoving, high z 387

luminosity 381

modulus 22

proper 380

Doppler

beaming 275

broadening 291, 295

shift 210, 217

Double-lobed extra-galactic radio

source 272

Cygnus A 272

Dust

absorption arc emission 109

carbon forms 110

classical grains 109

cometary 104

disks around stars 141

extinction efficiency factors 371

grain sites 110

grey 383

IR emission 109

meteoritic 104

molecular clouds relation 105

observational effects 105

origin of 111

very small grains (VSGs) 109

Dynamic range 51

Dyson sphere 4

Eddington luminosity 180

Einstein A coefficient 98, 254, 289, 349,

363

Einstein Ring 221, 224

Electric dipole

moment 281

radiation 358

Electromagnetic radiation xviii

useful equations xvix

Electromagnetic Spectrum, data 394

Electrons

CR spectrum 116

relativistic 255

Electrostatic units xxv

Emission coefficient 190, 235

thermal Bremsstrahlung synchrotron 266

Emission measure 240, 300, 303

Energy density 13

critical 384

geometry 14

specific 13

Energy levels 103

kinetic 187

of hydrogen 350–351

Energy levels

potential 187

rest mass 114

Epoch of reionization 170

Equation of radiative transfer 189, 191

LTE Solution 194

solutions of 191, 194

Equation of state 89

INDEX 409

Equation of transfer see Equation of radiative

transfer

Equilibrium

LTE 93

statistical 93, 96

thermal 86, 87

thermodynamic 93

timescale 86

Equipartition of energy 270

Excitation parameter 171, 244

Excited states populations 103

Extinction curve 108, 182

interstellar 108

Extinction efficiency factor 182,

369

Extinction 106, 153, 182, 369

selective 106

total 107

Eye

data 394

spectral response 29, 30

telescope analogy 32

False colour 52, 54

Field of view 37

Filter band passes 5

data 20

response functions 6

Fine structure 353

Flat field correction 49

Fluorescence 165, 262

Flux density 7

Flux 7

geometry 9

measurement 12, 13

Focal plane array 36

Focal ratio 33

Folded optics 34

Forbidden lines 201, 254, 353

Force

electrostatic xxiv

Lorentz 255, 358

strong 76

Fossils synchrotron 273

Fourier transform 238

Fraunhofer lines 165, 197

illustration 198

Free-bound emission 251–253

Free-free absorption 237

emission 237

processes 177

Friedmann Equation 383, 385

Fundamental frequency 259

Fusion, nuclear 71

Galactic Centre (GC) 326

nuclear Star cluster 328

radio images 326

Galaxies

precursors image 66

spectral plot 32

spectrum 31, 320

Gamma-ray bursts 155, 167

GRB 990123 166

Gamma-ray Spectrum,

of Galaxy 323

Gamma rays26Al – 288

in milky way 287

Spectral lines 286

Gases

characteristics of 82, 104

partially ionized 102

physical conditions of 83

Gaunt factor 238, 351

free-bound 252

plot 239

radio 243

x-ray 247

Gdwarf problem 81

Ghosts, synchrotron 273

Giant molecular clouds

311, 331

Global warming, plot 139

Globular clusters

formation epoch 335

NGC 6397 331, 334

Gravitational bending 209

Gravitational lenses 68, 220–227

and distances determination 226

characteristics 221

geometry 223

lens equation 223

mass mapping 222

Gravitational waves xx

Greek alphabet 389

410 INDEX

Greisen-Zatsepin-Kuz’min (GZK)

cutoff 118

Grey bodies 134, 140

geometry 139

Ground State 348

Guillotine factor 177

Gyrofrequency 258

relativistic 263

H I

21 cm line 307, 354

cold neutral medium 204

column density 309

emission/absorption illustration 203

expanding shell 102

in ISM 98, 101

in m 81 group 307

mass of 310

optical depth 200

shadowing 248

temperature determination geometry

202

temperature of 200

warm neutral medium 204

X-ray anti-correlations 246

H II regions 101, 160

free-free emission from 242

IC 5146 243

Logon Nebula 87

optical spectrum 253

properties from radio emission 244

ultra-compact 298

H2 molecule 282, 311

H ion 178

Harmonics 259, 264, 374

He flash 74

Heat engine, stellar 179

Heating 168

Heisenberg uncertainty Principle 73, 146

Hertzsprung-Russell (HR) diagram 23, 133

see also colour-magnitude diagram

Homogeneous 208

Hot Jupiters 141

Hubble constant 218

Hubble parameter 378

Hubble relation 218, 377

plot 379

Huygen’s Principal 38

Hydrogen

atomic data 349

Balmer lines 302

Balmer series 302, 350

energy level diagram 350

Lyman series 350

quantum data 351

spectrum 347

Hydrostatic equilibrium 250

Hypernovae 167

Ideal gas law xxvi, 88, 89

Ideal gas 88

Images

characteristics of 51

illustrations 53

visualization 52

Impact parameter 221, 238

Index of refraction, complex 182, 370 see

also Refraction index of

Initial mass function (IMF) 79, 81

plot 79

Instability strip 24, 180

Intensity 9

distance independence 13

flux relation 9, 11

geometry 11

mean 14

specific 9

Interstellar medium (ISM) 75, 104

Interstellar scintillation (see Scintillation

interstellar)

Intracluster gas 249, 251

Inverse Compton radiation (Scattering,

Inverse Compton)

synchrotron comparison 274

Io Plasma torus 260

Ion 84

Ionization equilibrium 170

Ionization 169

collisional 169

photo- 169

potential 169

state 82

Ionosphere 41, 376

Isotope 70

INDEX 411

Isotropic 208

radiation 8

universe 208

Jansky 7

Jets from AGN 8

Kappa mechanism 179

Kinematic distances 215

Kirchoff’s Law 193

Kramers’ Law 178

Lamb shift 362

Larmor formula 264, 238

Larmor precession 353

Laser guide star 47

Lens,

achromatic – see gravitational

lenses

diverging – see refraction, interstellar

telescopic 32

Light curves 77, 382

of microlens 226

Light echo, GRB 031203 156

Line broadening 290

collisional 295

FWHM 293

pressure 295

Stark 296

temperature limit 293

thermal 291, 292

Line profile 290

flat-topped 294

Gaussian 291

Lorentzian 295

Voigt 295

Line radiation 234, 279 (see also

bound-bound processor)

Line shape function 290

Local hot bubble (LHB) 248, 271

illustration 248

Local Standard of Rest (LSR) 211

Local thermodynamic equilibrium

(see equilibrium, LTE)

Lorentz factor 113, 263, 273

Lorentz profile 363

plot 364

Lorentz transformations 263

Luminosity 3, 7

flux relation 7

in waveband 5

intercepted 3

spectral 5

stellar 133

Ly a forest 161

Spectrum 162

Lyman a transition 254

Lyman series – see Hydrogen, Lyman series

MACHOs 68, 225

Magnetic bottleneck 261

Magnetic Bremstrahlung (see Synchrotron

radiation)

Magnetic dipole 260

Magnetic field strength

cyclotron emission 259

synchrotron emission 268

Magnetic fields 255

of Jupiter 260

random components 270

sample strengths 257

uniform components 270

Magnetic threads 326

Magnetized plasma 256

Magnetosphere 260

of Jupiter 262

Magnitudes 19

absolute 22

apparent 19

flux density relation 19

sample values 21

zero point values 21

Main sequence 72, 133

Masers 99

H2O 99

Mass absorption coefficient 175

Mass fraction 70

Mass

determination of 213

of intracluster gas 250

Massive compact halo objects

(see MACHOs) 68

Matter xvii

dark (see dark matter)

412 INDEX

missing 69

particulate 112

Maxwell Boltzmann velocity dist’n xxvi,

84, 86, 97

plot 85

Maxwellian dist’n xxvi

Mean free path 89

geometry 90

Mean molecular weight 88

Metallicity 70, 79, 81

Meteoritics

impact crater 1 xviii

micrometeorites 104

Metric 208

Robertson-Walker 208, 381

Schwarzschild 208, 219

Microlensing 225

Mie theory 182, 369, 372

Milky way

sketch 67

star formation rate 71

Minimum energy assumption 270

Mixing length theory 188

Modelling 253, 321–325

Molecular spectrum, of Orion

KL 284

Molecular transitions (see Transitions,

molecular) 281

Molecules H2 (see H2 molecule)

Molecules, CO (see CO molceule)

Molecules, detected 285

Monster 327

Natural line shape 365 (see also

Lorentz profile)

Nebulium 305

Neutron stars 77, (see also pulsars)

167

Noise 51

Non-Thermal emission 236

North Polar Spur 271

Nuclear burning 71

endothermic 77

exothermic 77

Nucleons 76

Nucleosynthesis

Big Bang 250

explosive 76, 287

H 71

He 73

primordial 65, 69

Number fraction 70

Observatories

LIGO xxi

SNO xxi

Olbers’

Paradox 15

Paradox, illustration 15

Opacity 175 (see also optical depth)

dynamics of 178

Opaque/transparent geometry 125

Optical depth 175

Optically thick 176

Optically thin 176

Oscillator model 359, 362

Oscillator strength 300, 362

Pacholczyk’s constants 266

values 266

Panchromatic observations 31

Partition function 97, 100

Pauli Exclusion Principal 73, 146, 353

Perfect gas law (see Ideal gas law)

Photoelectric effect 37, 279

Photon xviii

Physical constants 390

Pitch angle 255

Pixel 36

Pixel field of view 37

Planck function 124 (see also Blackbody

radiation) 146–148

Planck function, sample curves 124

Planetary data 393

Planetary nebulae 74

free-free emission from 245

images 75

Planets

extra solar 140–143

radio spectra 261

Plasma frequency 173, 374

Plasma waves 374

Plasmas 84, 373

diverging lens 174

perturbation 374

INDEX 413

Plate scale 33

Point spread function (PSF)

45

Polarizability 366

Polarization

by dust 372

cyclotron radiation 258

degree of 25

from scattering 156

geometry 360

of atmosphere 165

Synchrotron radiation 270

Polycyclic aromatic hydrocarbons

(PAHs) 110

examples 111

Population Synthesis 325

Power 3

spectral 5

P-P chain 71

Pressure

particle 88

radiation, geometry 18

total 89

radiation 16, 17

Principal quantum number 348

Probability dist’n function xxvi

Proper motion 211

Pulsars 8

anomalous X-ray 322

as ISM probes 174

Quantization 347

Quantum mechanics 279, 347

Quarks 76

Quasi-stellar objects (QSOs)

25

Quintessence 386

Radiation xvii, xviii, 188

Radio bursts 261

Radioactive decay 286

Random walk 95

illustration 96

Rayleigh criterion 40

Rayleigh-Jeans Law 129

Recombination – see free-bound emission

Recombination coefficient 170

effective 303

sample value 304

Recombination lines 102

optical 302

radio absorption 302

radio image 299

radio spectra 298

radio 297

Recombination radiation 251

Red giant 72

Reddening 306

atmospheric 48

interstellar 106

sunrise, set 164

Redshift 210

expansion 217

gravitational 210

Red super giant 74

Reflection nebula 154

Merope 155

Reflection 154

Refraction index of 58, 173, 370

atmospheric 42, 58

atmospheric, geometry 58

atmospheric, illustration 42

gravitational 220

interstellar 172

wavefront geometry 173

Refractive angle 59

data 61

plots 60

Relics 273

Resolution 33

diffraction-limited 38, 40

diffraction-limited, geometry

38

illustration 40

pixel-limited 45

seeing-limited 45

Resonance 361, 362, 376

Rosseland mean opacity 177

Rotation curve 216

Rotational transition (see Transition,

rotational) 281

Scale factor 219

Scattering 153, 357

414 INDEX

by dust 369

coherent 357

Compton 166, 367

elastic 157, 357

geometry 154, 155, 367

incoherent 165, 357

inelastic 165, 357

Inverse Compton 116, 168, 273

Rayleigh 362, 366

resonance 158, 361

Thomson 157, 177, 358

Schwarzschild Radius 220

Scintillation

atmospheric 48

interstellar 174

Seeing 43, 45

Self-absorption 194, 239

Sidelobes 39

Signal processing 49

Signal-to-noise ratio 51

Sky dips 50

Sky

blue colour 163

night colour 164

Snell’s law 42, 58, 172

Saha equation 100

Solar constant 8

variation 10

Solar eclipse 199

Solar flares 199

radio burst 375

X-ray Spectrum 200

Solar sail 18

Solar wind 117

Source function 191

Space weather 174

Space

2-D visualization 209

Euclidean 208, 380

Space-time 207, 210

Spallation 77, 113

Speckles 43

illustration 44

Spectral lines 5, 279

broadening (see Line broadening) 290

emission/absorption 196, 197

line strength 289

LTE 289

ratios 304

series 101

thermalization 289

Spectra

complex 320

modelling 321

Spectral band 283

Spectral index

CR energy 114

synchrotron 267

Spectral surveys 283

Spectral type stellar 104, 200

Spectrum 5

radio 324

Spiral galaxies

inclination 214

interior mass 216

rotation of 214

rotation or geometry 215

Standard candles 179, 382

Star formation 70

tracers of 80, 288

Star forming region, Cygnus 329

Stars

colours 131

dwarfs 72

high mass limit 181

low mass limit 71, 335

Statistical mechanics 86

Statistical weight 97, 146, 352

Stefan-Boltzmann Law 133

Stellar data,

dwarfs 395

giants 396

supergiants 397

Stellar evolution 70

1 m 73

low mass end products 74

Stellar pulsation 180

Stellar Spectra 201

Stellar winds 167, 180

Stimulated emission 191

Stromgren sphere 170

Sublimation 169

Sun

atmospheric conditions 198

INDEX 415

chromosphere 197

corona 84, 199

coronal loop 199, 257

granulation 188

granulation, illustration 189

interior model 391

photosphere 197

prominences 197, 256

Spectrum 127

Sunyaev-Zeldovich effect 275

(see also Inverse Compton radiation)

275

Superbubbles 271

Supermassive black hole (see Black holes,

supermassive) 181

Supernova explosion 75

Supernova remnants 5, 77, 117, 267

Cas A 5

edge-brightened 330

expanding 291

Supernovae 117, 155, 287

illustration 76

kinetic energy 75

SN 1987A 287

Type Ia 382

Type II 75

Symbols xxiii

Synchrotron sources, total energy of 269

Synchrotron radiation 116, 255

all-sky map 271

geometry 256

Source properties 268

spectra 267

spectral index 267

spectrum 262

Synchrotron self-absorption 266

Synchrotron sources 270

Telescopes 32

detectors 32, 36

detectors, focal plane arrays 37

geometry 33

ground-based Arecibo 35

ground-based Gemini 35, 46

ground-based, JCMT 42

objective 32

ray diagram 34

space-based Herschel 109

space-based, Hipparcos 23

space-based, GAIA 24

space-based, Chandra 36

spatial response 39

super-resolution 39

space based Spitzer 109

Temperature

brightness 127

dust 87

excitation 98

in keV 247

kinetic 84

‘negative’ 98

radiation 94

spin 99

TB-T relation 196

Theory of relativity

General xx, 207

Special 207

Thermal correction 49

Thermal emission 236

Thermonuclear reactions 71

Thomson cross-section 358 (see also

Scattering, Thomson)

Time dilation 210

Time variability & source size 227

Time

diffusion 96

look-back 380

Transitions

bound-bound 280

collisional 99, 289

collisionally dominated 94

electronic 280

electronic, physical conditions 296

forbidden 305, 354

molecular 281, 282, 311

nuclear 286

rotational 281

spontaneous 289

vibrational 281

Triple alpha process 74

Turbulence

atmospheric 43

line broadening 293

Turn-off point 333

416 INDEX

Two-photon radiation 253

Ultraluminous IR galaxy 137

Units

cgs xxii

cgs-SI conversion xxii

equivalent xxiii, xxv

prefixes xxv

SI xxii

Universe

accelerating 383

components of energy density (O)69

evolution of 65

gaseous 82

liquids in 82

liquids in, Titan 83

raisin cake model 378

Unresolved source 41

illustration 13

Variable stars 179

Velocity

components of 211

group 376

mean square 84

mean 85

most probable 85

phase 375

radial 210

Velocity parameter 292

root-mean-square 85

transverse 210

Vibrational transition (see Transitions,

vibrational) 281

Virialization 312

Virtual particles 208

Warm ionized medium (WIM) free-free

emission from 245

Wave electromagnetic xx

Wave-particle duality of light xviii

Weakly interacting massive particles

(WIMPs) 68

White dwarfs 74, 188, 382

Wien’s Displacement Law 131

Wien’s Law 129

Wolf-Rayet stars 113, 167

wind illustration 167

X factor for CO-H2 conversion 312

X-ray emiting gas, free-free emission

from 245, 251

X-ray halos 249

NGC 5746 250

X-ray-HI anti-correlation 246

X-rays, intracluster gas 251

Zeeman effect 296, 354

Zenith angle 42

INDEX 417

Figure 1.2

Figure 1.4

Figure 2.18

Figure 2.19

Figure 3.1

Figure 3.7

Figure 3.13

Figure 3.22

Figure 4.9

Figure 5.8

Figure 6.4

Figure 6.6

Figure 8.7

408 MHz

Cas A

Cen ANorth Polar Spur

Cygnus A

LMC

CrabGC

Figure 8.15

Figure 9.7

Figure 10.5

Astrophysics : Decoding the Cosmos

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Transcript of Astrophysics : Decoding the Cosmos